1. Introduction
Throughout this paper, 
$(R,\mathfrak {m})$ denotes an 
$n$-dimensional, analytically irreducible, Noetherian, local domain.
The Hilbert–Samuel multiplicity of an 
$\mathfrak {m}$-primary ideal 
$\mathfrak {a}\subset R$ is a fundamental invariant of the singularities of 
$\mathfrak {a}$ and satisfies various convexity properties such as Teissier's Minkowski inequality [Reference TeissierTei78]. In this paper, we consider 
$\mathfrak {m}$-filtrations of 
$R$, which generalize the filtrations of 
$R$ given by the powers of a single ideal, and prove various convexity properties for such filtrations. The results have applications to the study of K-stability, volumes of valuations, and problems in commutative algebra.
An 
$\mathfrak {m}$-filtration is a collection 
$\mathfrak {a}_\bullet =(\mathfrak {a}_\lambda )_{\lambda \in \mathbb {R}_{>0}}$ of 
$\mathfrak {m}$-primary ideals of 
$R$ that is decreasing, graded, and left continuous. The latter three conditions mean 
$\mathfrak {a}_\lambda \subset \mathfrak {a}_{\mu }$ when 
$\lambda >\mu$, 
$\mathfrak {a}_{\lambda }\cdot \mathfrak {a}_{\mu }\subset \mathfrak {a}_{\lambda +\mu }$, and 
$\mathfrak {a}_{\lambda }= \mathfrak {a}_{\lambda -\epsilon }$ when 
$0<\epsilon \ll 1$, respectively. The key examples of 
$\mathfrak {m}$-filtrations are as follows.
(1) A trivial example is given by taking powers
$( \mathfrak {b}^{\lceil \lambda \rceil })_{\lambda \in \mathbb {R}_{>0}}$ of a fixed $\mathfrak {m}$-primary ideal $\mathfrak {b}\subset R$.(2) An important example in this paper is
$\mathfrak {a}_{\bullet }(v):= (\mathfrak {a}_{\lambda }(v))_{\lambda \in \mathbb {R}_{>0}}$, where $v\colon {\rm Frac}(R)^\times \to \mathbb {R}$ is a valuation centered at $\mathfrak {m}$ and $\mathfrak {a}_{\lambda }(v) := \{f \in R\mid v(f) \geq \lambda \}$ (see § 2.2).(3) If
$(\mathfrak {b}_{\lambda })_{\lambda \in \mathbb {Z}_{>0}}$ is a decreasing, graded sequence of $\mathfrak {m}$-primary ideals,Footnote 1 then $(\mathfrak {b}_{\lceil \lambda \rceil })_{\lambda \in \mathbb {R}_{>0}}$ is an $\mathfrak {m}$-filtration. Such sequences have been well studied in the literature [Reference LazarsfeldLaz04, § 2.4.B].
Following work of Ein, Lazarsfeld, and Smith, the multiplicity of an 
$\mathfrak {m}$-filtration is
\[ \mathrm{vol}(\mathfrak{a}_\bullet) := \lim_{ \mathbb{N} \ni m\to\infty} \frac{\ell(R/\mathfrak{a}_m)}{m^n/n!} = \lim_{ \mathbb{N} \ni m\to\infty} \frac{\mathrm{e}(\mathfrak{a}_m)}{m^n} , \]
where the existence of the above limit and the equality were proven in increasing generality by [Reference Ein, Lazarsfeld and SmithELS03, Reference MustaMus02, Reference Lazarsfeld and MustaLM09, Reference CutkoskyCut13, Reference CutkoskyCut14]. This invariant is the local counterpart of the volume of a graded linear series of a line bundle and has been studied both in the context of commutative algebra [Reference Cutkosky, Sarkar and SrinivasanCSS19, Reference CutkoskyCut21] and recently in work of C. Li and others on the normalized volume of a valuation [Reference LiLi18, Reference Li, Liu and XuLLX20].
In this paper, we prove two properties of the multiplicity of 
$\mathfrak {m}$-filtrations. The first is a convexity result, which has applications to volumes of valuations and C. Li's normalized volume function. The second is a generalization of a classical theorem of Rees [Reference ReesRee61] from the setting of ideals to filtrations and results in a characterization of when the Minkowski inequality for filtrations is an equality.
1.1 Multiplicity and geodesics
Given two 
$\mathfrak {m}$-filtrations 
$\mathfrak {a}_{\bullet,0}$ and 
$\mathfrak {a}_{\bullet,1}$, we define a segment of 
$\mathfrak {m}$-filtrations 
$(\mathfrak {a}_{\bullet,t})_{t\in [0,1]}$ interpolating between 
$\mathfrak {a}_{\bullet,0}$ and 
$\mathfrak {a}_{\bullet,1}$ by setting
\[ \mathfrak{a}_{\lambda,t} := \sum_{\lambda=(1-t)\mu+t\nu} \mathfrak{a}_{\mu,0} \cap \mathfrak{a}_{\nu,1} . \]
We call 
$(\mathfrak {a}_{\bullet,t})_{t\in [0,1]}$ the geodesic between 
$\mathfrak {a}_{\bullet,0}$ and 
$\mathfrak {a}_{\bullet,1}$, since it is the local analogue of the geodesic between two filtrations of the section ring of a polarized variety [Reference Blum, Liu, Xu and ZhuangBLXZ23, Reference RebouletReb22]. This definition is also related to a construction in [Reference Xu and ZhuangXZ21].
In [Reference Blum, Liu, Xu and ZhuangBLXZ23], it was shown that several non-Archimedean functionals from the theory of K-stability are strictly convex along geodesics in the global setting. In a similar spirit, we prove a convexity result in the local setting for the multiplicity along geodesics.
Theorem 1.1 Assume 
$R$ contains a field. If 
$\mathfrak {a}_{\bullet,0}$ and 
$\mathfrak {a}_{\bullet,1}$ are 
$\mathfrak {m}$-filtrations with positive multiplicity, then the function 
$E(t) \colon [0,1]\to \mathbb {R}$ defined by 
$E(t) := \mathrm {e}(\mathfrak {a}_{\bullet,t})$ satisfies the following properties:
(1)
$E(t)$ is smooth;(2)
$E(t)^{-1/n}$ is concave, meaning
\[ E(t)^{-1/n}\ge (1-t)E(0)^{-1/n}+tE(1)^{-1/n} \quad \text{for all } t\in [0,1]; \]
(3)
$E(t)^{-1/n}$ is linear if and only if $\widetilde {\mathfrak {a}}_{\bullet,0} =\widetilde {\mathfrak {a}}_{c\bullet,1}$ for some $c\in \mathbb {R}_{>0}$.
The term ‘smooth’ in Theorem 1.1(1) means 
$E(t)$ extends to a 
$C^\infty$ function on 
$(-\epsilon,1+\epsilon )$ for some 
$\epsilon >0$. The symbol 
$\widetilde {\cdot }$ in Theorem 1.1(3) denotes the saturation of an 
$\mathfrak {m}$-filtration, which is defined in § 3.1. This notion is an analogue of the integral closure of an ideal in the setting of filtrations and discussed further in § 1.2 below.
The proof of Theorem 1.1 is inspired by a related argument in the global setting [Reference Blum, Liu, Xu and ZhuangBLXZ23] and relies on constructing a measure on 
$\mathbb {R}^2$ that encodes the multiplicities of the filtrations along the geodesic. In the special case when 
$\mathfrak {a}_{\bullet,0}$ is the 
$\mathfrak {m}$-filtration of a valuation minimizing the normalized volume function over a Kawamata log terminal (klt) singularity, the proof of [Reference Xu and ZhuangXZ21] can be used to show Theorem 1.1(2). Theorem 1.1 removes these strong restrictions and is proven without the theory of K-stability for valuations introduced in [Reference Xu and ZhuangXZ21, § 3.1].
1.1.1 Applications to volume
As first defined by Ein, Lazarsfeld, Smith, the volume of a valuation 
$v\colon {\rm Frac}(R)^{\times } \to \mathbb {R}$ centered at 
$\mathfrak {m}$ is
\[ \mathrm{vol}(v) := \mathrm{e}(\mathfrak{a}_\bullet(v)) = \lim_{m \to \infty} \frac{ \ell \big(R/ \mathfrak{a}_{m}(v)\big) }{m^n/n!} . \]
This invariant is a local analogue of the volume of a line bundle and also plays a role in the study of K-stability of Fano varieties and Fano cone singularities.
Theorem 1.1 can be applied to show that the volume of a valuation is strictly log convex on simplices of quasi-monomial valuations in the valuation space of 
$(R,\mathfrak {m})$. This gives an affirmative answer to [Reference Li, Liu and XuLLX20, Question 6.23]. We note that the volume was previously shown to be Lipschitz continuous on such a simplex [Reference Boucksom, Favre and JonssonBFJ14, Corollary D].
Corollary 1.2 (Convexity of volume)
Assume 
$R$ contains a field. Let 
$\eta \in (Y,D_1+\cdots +D_r)$ be a log smooth birational model of 
$(R,\mathfrak {m})$. For any 
$\boldsymbol {\alpha },\boldsymbol {\beta } \in \mathbb {R}_{>0}^{r}$ and 
$t\in (0,1)$,
\[ \mathrm{vol}(v_{(1-t)\boldsymbol {\alpha}+ t \boldsymbol {\beta}})^{-1/n} \geq (1-t)\,\mathrm{vol}(v_{\boldsymbol {\alpha}})^{-1/n} + t \,\mathrm{vol}(v_{\boldsymbol {\beta}})^{-1/n} \]
and equality holds if and only if 
$\boldsymbol {\alpha } = c \boldsymbol {\beta }$ for some 
$c\in \mathbb {R}_{>0}$.
In the above theorem, 
$v_{\boldsymbol {\alpha }}$ denotes the quasi-monomial valuation of 
${\rm Frac}(R)$ with weights 
$\boldsymbol {\alpha }$ on 
$D_1,\ldots, D_r$. See § 2.2.3 for a detailed definition.
1.1.2 Applications to normalized volume
In [Reference LiLi18], Chi Li defined the normalized volume of a valuation over a klt singularity and proposed the problem of studying its minimizer. The notion plays an important role in the study of K-stability of Fano varieties and in the study of klt singularities. The invariant has been extensively studied in the recent years; see [Reference Li, Liu and XuLLX20] and [Reference ZhuangZhu23] for surveys on this topic.
The fundamental problem in the study of the normalized volume function is the stable degeneration conjecture proposed by Li [Reference LiLi18, Conjecture 7.1] and Li and Xu [Reference Li and XuLX18, Conjecture 1.2]. The conjecture predicts that there exists a valuation minimizing the normalized volume function and that the minimizer is unique up to scaling, quasi-monomial, has finitely generated associated graded ring, and induces a degeneration of the klt singularity to a K-semistable Fano cone singularity. These five statements were proven in [Reference BlumBlu18, Reference Li and XuLX18, Reference XuXu20, Reference Xu and ZhuangXZ21, Reference Xu and ZhuangXZ22]. In particular, the stable degeneration conjecture is now a theorem.
Using Theorem 1.1, we revisit Xu and Zhuang's theorem stating that the minimizer of the normalized volume function is unique up to scaling [Reference Xu and ZhuangXZ21]. (The uniqueness was also previously proven in [Reference Li and XuLX18] under the assumption that the minimizer has finitely generated associated ring. The latter was recently shown in [Reference Xu and ZhuangXZ22].) In particular, we give a proof of the uniqueness result independent of the theory of K-semistability for valuations developed in [Reference Xu and ZhuangXZ21].
Corollary 1.3 (Uniqueness of minimizer)
If 
$x\in (X,D)$ is a klt singularity defined over an algebraically closed field of characteristic 
$0$, then any minimizer of 
$\widehat {\mathrm {vol}}_{X,D,x}$ (see Definition 2.8) is unique up to scaling.
The new proof of Corollary 1.3 takes the following direct approach. Fix two valuations 
$v_0$ and 
$v_1$ that minimize 
$\widehat {\mathrm {vol}}_{X,D,x}$ and consider the geodesic 
$(\mathfrak {a}_{\bullet,t})_{t\in [0,1]}$ between 
$\mathfrak {a}_{\bullet }(v_0)$ and 
$\mathfrak {a}_{\bullet }(v_1)$. Using Theorem 1.1(2), a characterization of the infimum of the normalized volume function in terms of normalized multiplicities [Reference LiuLiu18, Theorem 27], and an inequality of log canonical thresholds [Reference Xu and ZhuangXZ21, Theorem 3.11], we show that 
$\mathrm {e}(\mathfrak {a}_{\bullet,t})$ is linear. Theorem 1.1(3) then implies 
$cv_0=v_1$ for some 
$c>0$.
1.2 Rees's theorem
A theorem of Rees [Reference ReesRee61] states that if 
$\mathfrak {a}\subset \mathfrak {b}$ are two 
$\mathfrak {m}$-primary ideals, then the following statements are equivalent.
(1)
$\mathrm {e}(\mathfrak {a})=\mathrm {e}(\mathfrak {b})$.(2)
$\overline { \bigoplus _{m\in \mathbb {N}} \mathfrak {a}^m}=\overline { \bigoplus _{m \in \mathbb {N}} \mathfrak {b}^m}$.(3)
$\overline {\mathfrak {a}}=\overline {\mathfrak {b}}$.
The symbol 
$\overline {\, \cdot \, }$ in (2) denotes the algebraic closure in 
$R[t]$, while in (3) it denotes the integral closure of an ideal. The equivalence between (2) and (3) follows from definitions.
It is natural to ask for a generalization of the above result for 
$\mathfrak {m}$-filtrations 
$\mathfrak {a}_\bullet \subset \mathfrak {b}_\bullet$. In [Reference CutkoskyCut21], Cutkosky studies whether the two conditions
(1)
$\mathrm {e}(\mathfrak {a}_\bullet )=\mathrm {e}(\mathfrak {b}_\bullet )$,(2)
$\overline {\bigoplus _{m\in \mathbb {N}} \mathfrak {a}_m}=\overline { \bigoplus _{m \in \mathbb {N}} \mathfrak {b}_m}$
are equivalent. While (2) 
$\Rightarrow$ (1) holds by [Reference Cutkosky, Sarkar and SrinivasanCSS19, Theorem 6.9], (1) 
$\Rightarrow$ (2) can fail even in very simple examples (see Example 3.7). That said, (1) 
$\Rightarrow$ (2) holds for special classes of 
$\mathfrak {m}$-filtrations [Reference CutkoskyCut21, Theorem 1.4].
To remedy this issue, we introduce the saturation 
$\widetilde {\mathfrak {a}}_\bullet$ of an 
$\mathfrak {m}$-filtration 
$\mathfrak {a}_\bullet$ in § 3. (The definition may be viewed as a local analogue of a construction studied by Boucksom and Jonsson in [Reference Boucksom and JonssonBJ21]; see § 6.) The saturation is defined using divisorial valuations, analogous to the valuative definition of the integral closure of an ideal. Using this notion, we prove a version of Rees's theorem for filtrations.
Theorem 1.4 (Rees's theorem)
For 
$\mathfrak {m}$-filtrations 
$\mathfrak {a}_\bullet \subset \mathfrak {b}_\bullet$, 
$\mathrm {e}(\mathfrak {a}_\bullet )=\mathrm {e}(\mathfrak {b}_\bullet )$ if and only if 
$\widetilde {\mathfrak {a}}_\bullet =\widetilde {\mathfrak {b}}_\bullet$.
The above result can be explained as follows. The multiplicity of an 
$\mathfrak {m}$-filtration is determined by its valuative properties, not by the integral properties of its Rees algebra. These two properties coincide for ideals, but do not always coincide for filtrations by Example 3.7.
1.3 Minkowski inequality
By work of Teissier [Reference TeissierTei78], Rees and Sharp [Reference Rees and SharpRS78], and Katz [Reference KatzKat88] in increasing generality, for two 
$\mathfrak {m}$-primary ideals 
$\mathfrak {a}$ and 
$\mathfrak {b}$ of 
$R$,
\[ \mathrm{e}(\mathfrak{a}\mathfrak{b})^{1/n}\leq \mathrm{e}(\mathfrak{a})^{1/n}+\mathrm{e}(\mathfrak{b})^{1/n} \]
and the equality holds if and only if there exist 
$c,d\in \mathbb {Z}_{>0}$ such that 
$\overline {\mathfrak {a}^c}=\overline {\mathfrak {b}^d}$.
For two 
$\mathfrak {m}$-filtrations 
$\mathfrak {a}_\bullet$ and 
$\mathfrak {b}_\bullet$, we let 
$\mathfrak {a}_\bullet \mathfrak {b}_\bullet$ denote the 
$\mathfrak {m}$-filtration 
$( \mathfrak {a}_\lambda \mathfrak {b}_\lambda )_{\lambda \in \mathbb {R}_{>0}}$. Using the saturation of a filtration, we characterize when the Minkowski inequality for filtrations is an equality.
Corollary 1.5 (Minkowski inequality)
For 
$\mathfrak {m}$-filtrations 
$\mathfrak {a}_\bullet$ and 
$\mathfrak {b}_\bullet$ with positive multiplicity,
\begin{equation} \mathrm{e}(\mathfrak{a}_\bullet\mathfrak{b}_\bullet)^{1/n}\leq \mathrm{e}(\mathfrak{a}_\bullet)^{1/n}+\mathrm{e}(\mathfrak{b}_\bullet)^{1/n} \end{equation}
and the equality holds if and only if 
$\widetilde {\mathfrak {a}}_{\bullet } = \widetilde {\mathfrak {b}}_{c\bullet }$ for some 
$c\in \mathbb {R}_{>0}$.
The inequality statement of the above corollary is not new and is due to Musta [Reference MustaMus02], Kaveh and Khovanskii [Reference Kaveh and KhovanskiiKK14], and Cutkosky [Reference CutkoskyCut15] in increasing levels of generality. In the equality statement, the forward implication follows easily from [Reference CutkoskyCut21, Theorem 10.3] and the definition of the saturation, while the reverse implication relies on Theorem 1.4.
Remark 1.6 (Relation to work of Cutkosky)
Cutkosky proved a version of the equality part of Corollary 1.5 in the special case when 
$\mathfrak {a}_\bullet$ and 
$\mathfrak {b}_\bullet$ are bounded filtrations [Reference CutkoskyCut21, Definition 1.3], which roughly means that their integral closure is induced by a finite collection of divisorial valuations [Reference CutkoskyCut21, Theorem 1.6]. For such filtrations, the integral closure and saturation agree by Lemma 3.20. Thus, Corollary 1.5 may be viewed as a generalization of the latter result. Similarly, Theorem 1.4 may be viewed as a generalization of [Reference CutkoskyCut21, Theorem 1.4].
This paper is organized as follows: In § 2, we recall definitions and basic facts concerning filtrations, valuations, and multiplicities. In § 3, we introduce the saturation of a filtration and prove Theorem 1.4. In § 4, we define the geodesic between two filtrations and prove Theorem 1.1. In § 5, we deduce Corollaries 1.2, 1.3, and 1.5 as consequences of results in the previous two sections. In § 6, we discuss relations between the results in this paper and global results in the K-stability literature. The appendix of the paper is devoted to an alternate proof of a special case of Theorem 1.1 using the theory of Okounkov bodies.
2. Preliminaries
Throughout this section, 
$(R,\mathfrak {m},\kappa )$ denotes an 
$n$-dimensional, analytically irreducible,Footnote 2 Noetherian, local domain. We set 
$X:= \mathrm {Spec}(R)$ and write 
$x\in X$ for the closed point corresponding to 
$\mathfrak {m}$.
2.1 Filtrations
Definition 2.1 An 
$\mathfrak {m}$-filtration is a collection 
$\mathfrak {a}_\bullet =(\mathfrak {a}_\lambda )_{\lambda \in \mathbb {R}_{>0}}$ of 
$\mathfrak {m}$-primary ideals of 
$R$ such that
(1)
$\mathfrak {a}_\lambda \subset \mathfrak {a}_{\mu }$ when $\lambda >\mu$,(2)
$\mathfrak {a}_{\lambda } = \mathfrak {a}_{\lambda -\epsilon }$ when $0<\epsilon \ll 1$, and(3)
$\mathfrak {a}_{\lambda } \cdot \mathfrak {a}_{\mu } \subset \mathfrak {a}_{\lambda + \mu }$ for any $\lambda,\mu \in \mathbb {R}_{>0}$.
By convention, we set 
$\mathfrak {a}_{0}:= R$. The definition is a local analogue of a filtration of the section ring of a polarized variety in [Reference Boucksom, Hisamoto and JonssonBHJ17].
For 
$\lambda \in \mathbb {R}_{\ge 0}$, define 
$\mathfrak {a}_{>\lambda }:=\bigcup _{\mu >\lambda }\mathfrak {a}_\mu$. If 
$\lambda$ satisfies 
$\mathfrak {a}_{>\lambda }\subsetneq \mathfrak {a}_\lambda$, then we say 
$\lambda$ is a jumping number of the filtration.
The scaling of an 
$\mathfrak {m}$-filtration 
$\mathfrak {a}_\bullet$ by 
$c\in \mathbb {R}_{>0}$ is 
$\mathfrak {a}_{c\bullet }:=(\mathfrak {a}_{c \lambda } )_{\lambda \in \mathbb {R}_{>0}}$. The product of two 
$\mathfrak {m}$-filtrations 
$\mathfrak {a}_\bullet$ and 
$\mathfrak {b}_\bullet$ is 
$\mathfrak {a}_\bullet \mathfrak {b}_\bullet :=(\mathfrak {a}_\lambda \mathfrak {b}_\lambda )_{\lambda \in \mathbb {R}_{>0}}$. Both are again 
$\mathfrak {m}$-filtrations.
Definition 2.2 An 
$\mathfrak {m}$-filtration 
$\mathfrak {a}_\bullet$ is linearly bounded if there exists a constant 
$c>0$ such that 
$\mathfrak {a}_{\lambda } \subset \mathfrak {m}^{\lceil c \lambda \rceil }$ for all 
$\lambda \in \mathbb {R}_{>0}$.
Lemma 2.3 Let 
$\mathfrak {a}_\bullet$ and 
$\mathfrak {b}_\bullet$ be 
$\mathfrak {m}$-filtrations. If 
$\mathfrak {a}_\bullet$ is linearly bounded, then there exists 
$c\in \mathbb {R}_{>0}$ such that 
$\mathfrak {a}_{c\lambda } \subset \mathfrak {b}_\lambda$ for all real numbers 
$\lambda \geq 1$
Proof. Since 
$\mathfrak {a}_\bullet$ is linearly bounded, there exists 
$c_0 \in \mathbb {R}_{>0}$ such that 
$\mathfrak {a}_{\lambda } \subset \mathfrak {m}^{ \lceil c_0\lambda \rceil }$ for all 
$\lambda >0$. Since 
$\mathfrak {b}_1$ is an 
$\mathfrak {m}$-primary ideal, there exists 
$d\in \mathbb {Z}_{>0}$ such that 
$\mathfrak {m}^{d} \subset \mathfrak {b}_1$. If we set 
$c:= 2d/c_0$, then
\[ \mathfrak{a}_{ c\lambda } \subset \mathfrak{m}^{\lceil 2d \lambda \rceil } \subset \mathfrak{m}^{d \lceil \lambda \rceil } \subset (\mathfrak{b}_1)^{\lceil \lambda \rceil} \subset \mathfrak{b}_{\lceil \lambda \rceil } \subset \mathfrak{b}_{\lambda} \]
for all 
$\lambda \geq 1$. (The second inclusion uses that 
$d\lceil \lambda \rceil \leq d(\lambda +1) \leq 2d \lambda \leq \lceil 2 d \lambda \rceil$).
2.2 Valuations
Let 
$(\Gamma,\ge )$ be a totally ordered abelian group. A 
$\Gamma$-valuation of 
$R$ is a map 
$v\colon {\rm Frac(R)}^\times \to \Gamma$ such that
(1)
$v(fg)=v(f)+v(g)$, and(2)
$v(f+g)\ge \min \{v(f),v(g)\}$.
By convention, we set 
$v(0):= \infty$. We say 
$v$ is centered at 
$\mathfrak {m}$ if 
$v\geq 0$ on 
$R$ and 
$v>0$ on 
$\mathfrak {m}\subset R$.
A valuation 
$v$ of 
$R$ induces a valuation ring 
$R_{v} : = \{ f \in K \mid v(f) \geq 0\}$. We write 
$\mathfrak {m}_v$ for the maximal ideal of 
$R_v$ and 
$\kappa _{v}:=R_v/\mathfrak {m}_v$.
2.2.1 Real valuations
When 
$\Gamma =\mathbb {R}$ with the usual order, we say that 
$v$ is a real valuation. We denote by 
$\mathrm {Val}_{R,\mathfrak {m}}$ the set of real valuations centered at 
$\mathfrak {m}$.Footnote 3 In geometric settings, we will instead denote the set by 
$\mathrm {Val}_{X,x}$.
For 
$v\in \mathrm {Val}_{R,\mathfrak {m}}$ and 
$\lambda \in \mathbb {R}_{>0}$, we define the valuation ideal
\[ \mathfrak{a}_{\lambda}(v) := \{ f \in R \mid v(f) \geq \lambda \} \]
for each 
$\lambda \in R$. Using (1) and (2), one can show 
$\mathfrak {a}_\bullet (v)$ is an 
$\mathfrak {m}$-filtration.
For 
$v\in \mathrm {Val}_{R,\mathfrak {m}}$ and an ideal 
$\mathfrak {a}\subset R$, set 
$v(\mathfrak {a}):= \min \{v(f)\mid f\in \mathfrak {a}\}$. For an 
$\mathfrak {m}$-filtration 
$\mathfrak {a}_\bullet$, set
\[ v(\mathfrak{a}_\bullet):= \lim_{ \mathbb{N}\in m \to \infty} \frac{ v(\mathfrak{a}_m)}{m} = \inf_{m\in \mathbb{N}} \frac{v(\mathfrak{a}_m)}{m} , \]
where the existence of the limit and second equality is [Reference Jonsson and MustaţăJM12, Proposition 2.3].
2.2.2 Divisorial valuations
A valuation 
$v\in \mathrm {Val}_{R,\mathfrak {m}}$ is divisorial if
\[ {\rm tr.deg}_{\kappa}(\kappa_v) =n-1. \]
We write 
$\mathrm {DivVal}_{R,\mathfrak {m}}\subset \mathrm {Val}_{R,\mathfrak {m}}$ for the set of such valuations.
Divisorial valuations appear geometrically. If 
$\mu :Y\to X$ is a proper birational morphism with 
$Y$ normal and 
$E\subset Y$ a prime divisor, then there is an induced valuation 
$\mathrm {ord}_{E} \colon {\rm Frac}(R)^\times \to \mathbb {Z}$. If 
$\mu (E) =x$ and 
$c\in \mathbb {R}_{>0}$, then 
$c\cdot \mathrm {ord}_E \in \mathrm {DivVal}_{R,\mathfrak {m}}$. When 
$R$ is excellent, all divisorial valuations are of this form; see, for example, [Reference Cutkosky and SarkarCS22, Lemma 6.5].
2.2.3 Quasi-monomial valuations
In the following construction, we always assume 
$R$ contains a field. Let 
$\mu \colon Y:= \mathrm {Spec}(S) \to X=\mathrm {Spec}(R)$ be a birational morphism with 
$R\to S$ finite type and 
$\eta \in Y$ a not necessarily closed point such that 
$\mathcal {O}_{Y,\eta }$ is regular and 
$\mu (\eta ) =x$. Given a regular system of parameters 
$y_1,\ldots, y_r$ of 
$\mathcal {O}_{Y,\eta }$ and 
$\boldsymbol {\alpha }=(\alpha _1,\ldots, \alpha _r)\in \mathbb {R}_{\geq 0}^r\setminus {\bf 0}$, we define a valuation 
$v_{\bf \alpha }$ as follows. For 
$0\neq f\in \mathcal {O}_{Y,\eta }$, we can write 
$f$ in 
$\widehat {\mathcal {O}}_{Y,\eta } \simeq k(\eta ) [[y_1,\ldots, y_r]]$ as 
$\sum _{\boldsymbol {\beta }\in \mathbb {Z}_{\geq 0}^r} c_{\boldsymbol {\beta }} {y}^{\boldsymbol {\beta }}$ and set
\[ v_{\boldsymbol{\alpha}}(f):= \min \{ \langle \boldsymbol {\alpha}, {\boldsymbol{\beta}} \rangle \mid c_{\boldsymbol{\beta}} \neq 0 \}. \]
A valuation of the above form is called quasi-monomial.
Let 
$D= D_{1}+\cdots +D_r$ be a reduced divisor on 
$Y$ such that 
$y_i=0$ locally defines 
$D_i$ and 
$\mu (D_i)= x$ for each 
$i$. We call 
$\eta \in (Y,D)$ a log smooth birational model of 
$X$. We write 
$\mathrm {QM}_\eta (Y,E) \subset \mathrm {Val}_{X,x}$ for the set of quasi-monomial valuations that can be described at 
$\eta$ with respect to 
$y_1,\ldots, y_r$ and note that 
$\mathrm {QM}_\eta (Y,D) \simeq \mathbb {R}^r_{\geq 0} \setminus \boldsymbol {0}$.
2.2.4 Izumi's inequality
The order function 
$R\setminus 0 \to \mathbb {N}$ is defined by
\[ \mathrm{ord}_{\mathfrak{m}}(f):= \max \{ k \in \mathbb{Z}_{\geq 0} \mid f\in \mathfrak{m}^k \}. \]
The following version of Izumi's inequality compares 
$\mathrm {ord}_{\mathfrak {m}}$ to a fixed quasi-monomial valuation.
Lemma 2.4 Let 
$v\in \mathrm {Val}_{R,\mathfrak {m}}$. If (i) 
$v$ is divisorial or (ii) 
$R$ contains a field and 
$v$ is quasi-monomial, then there exists a constant 
$c>0$ such that
\[ v(\mathfrak{m}) \cdot \mathrm{ord}_\mathfrak{m}(f)\leq v(f)\leq c\cdot\mathrm{ord}_\mathfrak{m}(f) \]
for all 
$f\in R$. In particular, 
$\mathfrak {a}_\bullet (v)$ is linearly bounded.
Proof. Since 
$f \in \mathfrak {m}^{\mathrm {ord}_{\mathfrak {m}}(f)}$ by definition, 
$v(\mathfrak {m}) \cdot \mathrm {ord}_{\mathfrak {m}}(f) = v(\mathfrak {m}^{\mathrm {ord}_{\mathfrak {m}}(f)}) \leq v(f)$. It remains to prove the existence of 
$c>0$ such that 
$v(f) \leq c \cdot \mathrm {ord}_{\mathfrak {m}}(f)$ for all 
$f\in R$.
If (i) holds, the existence of 
$c$ follows from Izumi's theorem for divisorial valuations as phrased in [Reference Rond and SpivakovskyRS14, Remark 1.6].Footnote 4 If (ii) holds, then there exists a log smooth birational model 
$\eta \in (Y,D)$ of 
$x\in X$ and 
$\boldsymbol {\alpha } \in \mathbb {R}^r$ such that 
$v=v_{\boldsymbol {\alpha }}$. Choose 
$\boldsymbol {\gamma } \in \mathbb {Z}_{>0}^r$ such that 
$\boldsymbol {\alpha }_i \leq \boldsymbol {\gamma }_i$ for each 
$i=1,\ldots, r$ and consider the valuation 
$w: = v_{\boldsymbol {\gamma }}$. We claim that 
$w\in \mathrm {DivVal}_{R,\mathfrak {m}}$. Assuming the claim, then (i) implies that there exists 
$c>0$ such that 
$w(f) \leq c \cdot \mathrm {ord}_{\mathfrak {m}}(f)$ for all 
$f\in R$. Since 
$v(f)\leq w(f)$ for all 
$f\in R$, (ii) then follows.
To verify that 
$w\in \mathrm {DivVal}_{R,\mathfrak {m}}$, note that 
$w: {\rm Frac\ }(R)^\times \to \mathbb {R}$ is the composition
\[ {\rm Frac}(R)^\times \hookrightarrow {\rm Frac}( \widehat{\mathcal{O}}_{Y,\eta})^\times \overset{\widehat{w}}{\longrightarrow } \mathbb{R}, \]
where 
$\widehat {w}$ is the valuation that sends 
$\sum _{\beta \in \mathbb {Z}^r_{\geq 0} } c_{\boldsymbol {\beta }} y^{\boldsymbol {\beta }}$ to 
$\min \{ \langle \boldsymbol {\alpha }, \boldsymbol {\beta } \rangle \mid c_{\boldsymbol {\beta }} \neq 0\}$. Using that 
$w$ is 
$\mathbb {Z}$-valued, a computation (see, for example, [Reference Jonsson and MustaţăJM12, Proposition 3.7]) shows 
$\widehat {w}$ is a divisorial valuation of 
$\widehat {\mathcal {O}}_{Y,\eta }$. Hence, 
$w$ is a divisorial valuation of 
$\mathcal {O}_{Y,\eta }$ by [Reference Huneke and SwansonHS06, Proposition 9.3.5]. Now, we compute
\[ {\rm tr.deg}_{\kappa}(\kappa_{w}) = {\rm tr.deg}_{\kappa}(k(\eta)) + {\rm tr.deg}_{k(\eta)}(\kappa_v) = {\rm tr.deg}_{\kappa}(k(\eta)) + (\dim \mathcal{O}_{Y,\eta} -1) = \dim R. \]
To see that the last equality holds, note that the complete local ring 
$\widehat {R}$ is a domain by assumption and thus equidimensional. Therefore 
$R$ is universally catenary by [Sta, Tag 0AW6] and so the dimension formula [Sta, Tag 02II] gives the last equality. Therefore, 
$w$ is divisorial as desired.
2.3 Intersection numbers
The theory of intersection numbers of line bundles on a proper scheme over an algebraically closed fields was developed in [Reference KleimanKle66]. We will use a more general framework developed in [Reference KleimanKle05, Appendix B].
2.3.1 Definition
Let 
$Z$ be a proper scheme over an Artinian ring 
$\Lambda$. For line bundles 
$\mathcal {L}_1,\ldots, \mathcal {L}_r$ on 
$Z$, the function
\[ \mathbb{Z}^r \ni (m_1,\ldots, m_r ) \to \sum_{i\geq0 } (-1)^i \ell_{\Lambda} \big( H^i \big(Z, \mathcal{L}_1^{m_1} \otimes \cdots \otimes \mathcal{L}_r^{m_r}\big) \big) \in \mathbb{Z} \]
is a polynomial of degree 
$\leq \dim Z$ [Reference KleimanKle05, Theorem B.7]. The intersection number 
$(\mathcal {L}_1 \cdot \cdots \mathcal {L}_r)$ is defined to be the coefficient of 
$m_1\cdots m_r$ in the above polynomial. When the choice of 
$\Lambda$ is unclear, we will write 
${(\mathcal {L}_1 \cdot \ldots \cdot \mathcal {L}_r)}_\Lambda$.
2.3.2 Intersections of exceptional divisors
Let 
$\mu :Y\to X=\mathrm {Spec}(R)$ be a proper birational morphism with 
$Y$ normal. For Cartier divisors 
$F_1,\ldots, F_{n-1}$ on 
$Y$ and a Weil divisor 
$D: = \sum _{i=1}^r a_i D_i$ on 
$Y$ with support contained in 
$Y_\kappa$, we set
\[ F_1 \cdot \ldots \cdot F_{n-1} \cdot D :=\sum_{i=1}^r a_i (\mathcal{O}_{Y}(F_1)\vert_{D_i} \cdot \ldots \cdot \mathcal{O}_{Y}(F_{n-1})\vert_{D_i})_\kappa . \]
This is well defined, since each prime divisor 
$D_i$ is proper over 
$\mathrm {Spec}(\kappa )$.
Proposition 2.5 Assume 
$R$ is complete. If 
$F_1, \ldots, F_n$ are Cartier divisors on 
$Y$ with support contained in 
$Y_\kappa$, then 
$F_1\cdot \ldots \cdot F_n$ is independent of the ordering of the 
$F_i$.
Since the intersection product in § 2.3.1 is symmetric, 
$F_1\cdot \ldots \cdot F_n$ is independent of the order of 
$F_1,\ldots, F_{n-1}$. To deduce the full result, we rely on intersection theory [Reference FultonFul98].
A subtle issue is that the results in [Reference FultonFul98, § 1-18] are stated for schemes of finite type over a field and, hence, do not immediately apply to 
$Y$. Fortunately, the Chow group of a scheme of finite type over a regular base scheme can be defined and the results of [Reference FultonFul98, § 2] extend to this setting by [Reference FultonFul98, §20.1] (see also [Sta, Chapter 02P3] for the results in an even more general setting).
Proof. By Cohen's structure theorem, there exists a surjective map 
$A \twoheadrightarrow R$, where 
$A$ is a regular local ring. Since the composition 
$Y\to X:= \mathrm {Spec}(R)\to \mathrm {Spec}(A)$ is finite type, the framework of [Reference FultonFul98, § 20.1] applies. Using intersection theory on 
$Y$ and its subschemes, we compute
\begin{align*} F_1 \cdot \ldots \cdot F_n &= \sum_{i=1}^r a_i \int_{D_i} c_1(\mathcal{M}_1) \cap \cdots \cap c_1(\mathcal{M}_{n-1}) \cap [D_i]\\ &= \int_{Y_\kappa } c_1(\mathcal{M}_1) \cap \cdots \cap c_1(\mathcal{M}_{n-1}) \cap [F_n] = \int_{Y_\kappa} F_1 \cdot \ldots \cdot F_{n-1} \cdot [F_n], \end{align*}
where 
$\int _{D_i}$ and 
$\int _{Y_\kappa }$ denote the degree maps induced by the proper morphisms 
$D_i \to \mathrm {Spec}(\kappa )$ and 
$Y_\kappa \to \mathrm {Spec}(\kappa )$ [Reference FultonFul98, Definition 1.4] and 
$\mathcal {M}_i : = \mathcal {O}_Y(F_i)$. The first equality holds by [Reference FultonFul98, Example 18.3.6] with the fact that 
$D_i$ is a proper scheme over 
$\kappa$, the second by the equality 
$[F_n] = \sum _{i=1}^r a_i [D_i]$, and the last by the definition of the first Chern class. Since
\[ F_1 \cdot \ldots \cdot F_{n-1} \cdot [F_n] = F_1 \cdot \ldots \cdot F_n \cdot [Y] \in { A}_*(Y_\kappa) \]
and the latter is independent of the order of the 
$F_i$ by [Reference FultonFul98, Corollary 2.4.2], the proposition holds.
2.4 Multiplicity
2.4.1 Multiplicity of an ideal
The multiplicity of an 
$\mathfrak {m}$-primary ideal 
$\mathfrak {a}$ is
\[ \mathrm{e}(\mathfrak{a}):= \lim_{m\to \infty} \frac{\ell(R/\mathfrak{a}^m)}{m^n/n!} . \]
The following intersection formula for multiplicities commonly appears in the literature when 
$x\in X$ is a closed point on a quasi-projective variety [Reference LazarsfeldLaz04, p. 92]. In the generality stated below, it follows from [Reference RamanujamRam73].
Proposition 2.6 Let 
$\mathfrak {a}\subset R$ be an 
$\mathfrak {m}$-primary ideal and 
$Y\to \mathrm {Spec}(R)$ a proper birational morphism with 
$Y$ normal. If 
$\mathfrak {a} \cdot \mathcal {O}_Y=\mathcal {O}_Y(-E)$ for a Cartier divisor 
$E = \sum _i a_i D_i$, then
\[ \mathrm{e}(\mathfrak{a})= (-E)^{n-1} \cdot E . \]
Proof. By [Reference RamanujamRam73, Theorem and Remark 1], 
$\mathrm {e}(\mathfrak {a}) = (\mathcal {L}^{n-1})_{R/\mathfrak {a}}$, where 
$\mathcal {L}:= \mathcal {O}_Y(-E)\vert _E$. Observe that
\[ (\mathcal{L}^{n-1})_{R/\mathfrak{a}} = \sum \ell_{\mathcal{O}_{E,\xi_i}}( \mathcal{O}_{E,\xi_i}) (\mathcal{L}_i^{n-1})_{R/\mathfrak{a}} =\sum a_i (\mathcal{L}_i^{n-1})_{R/\mathfrak{a}} , \]
where 
$E= \sum a_i D_i$ and 
$\mathcal {L}_{i} := \mathcal {L}\vert _{E_i}$. Indeed, the first equality is [Reference KleimanKle05, Lemma B.12] and the second equality holds, since 
$\mathcal {O}_{Y,\xi _i}$ is a DVR and 
$\mathcal {O}_{E,\xi _i} = \mathcal {O}_{Y,\xi }/ (\pi ^{a_i})$, where 
$\pi$ is a uniformizer of the DVR.
To finish the proof, notice that 
$D_i$ is defined over 
$\kappa$, since there is a natural inclusion 
$D_i \hookrightarrow Y_{\kappa,\mathrm {red}} \to Y_{\kappa }$. Since the residue field of the local ring 
$R/\mathfrak {a}$ is 
$\kappa$, any finite-length 
$\kappa$-module 
$M$ satisfies 
$\ell _{R/\mathfrak {a}}(M) = \ell _{\kappa }(M)$ by [Sta, Lemma 02M0]. Thus,
\[ \sum_{i=1}^r a_i (\mathcal{L}_i^{n-1})_{R/\mathfrak{a}} = \sum_{i=1}^r a_i (\mathcal{L}_i^{n-1})_{\kappa} = (-E)^{n-1} \cdot E , \]
which completes the proof.
2.4.2 Multiplicity of a filtration
Following [Reference Ein, Lazarsfeld and SmithELS03], the multiplicity of a graded sequence of 
$\mathfrak {m}$-primary ideals 
$\mathfrak {a}_\bullet$ is
\[ \mathrm{e}(\mathfrak{a}_\bullet):= \lim_{m\to\infty}\frac{\ell(R/\mathfrak{a}_m)}{m^n/n!} \in [0,\infty). \]
By [Reference Ein, Lazarsfeld and SmithELS03, Reference MustaMus02, Reference Lazarsfeld and MustaLM09, Reference CutkoskyCut13, Reference CutkoskyCut14] in increasing generality, the above limit exists and
\begin{equation} \mathrm{e}(\mathfrak{a}_\bullet)=\lim_{m\to\infty}\frac{\mathrm{e}(\mathfrak{a}_m)}{m^n} = \inf_{m} \frac{\mathrm{e}(\mathfrak{a}_m)}{m^n}; \end{equation}
see, in particular, [Reference CutkoskyCut14, Theorem 6.5]. Also defined in [Reference Ein, Lazarsfeld and SmithELS03], the volume of a valuation 
$v\in \mathrm {Val}_{R,\mathfrak {m}}$ is
\[ \mathrm{vol}(v):= \mathrm{e}(\mathfrak{a}_\bullet(v)). \]
We set
\[ \mathrm{Val}_{R,\mathfrak{m}}^+:= \{v\in\mathrm{Val}_{R,\mathfrak{m}}\mid \mathrm{vol}(v)>0\}\subset \mathrm{Val}_{R,\mathfrak{m}} . \]
Proposition 2.7 Let 
$v\in \mathrm {Val}_{R,\mathfrak {m}}$. If (i) 
$v$ is divisorial or (ii) 
$R$ contains a field and 
$v$ is quasi-monomial, then 
$\mathrm {vol}(v)>0$.
Proof. By Lemma 2.4, there exists 
$c>0$ such that 
$\mathfrak {a}_{\bullet }(v) \subset \mathfrak {m}^{c\bullet }$. Thus, 
$\mathrm {vol}(v)\geq \mathrm {e}(\mathfrak {m}^{c\bullet }) = c^{-n}\mathrm {e}(\mathfrak {m})>0$.
2.5 Normalized volume
Assume 
$R$ is the local ring of a closed point on a algebraic variety defined over an algebraically closed field of characteristic 0. We say 
$x\in (X,\Delta )$ is a klt singularity if 
$\Delta$ is an 
$\mathbb {R}$-divisor on 
$X$ such that 
$K_{X}+\Delta$ is 
$\mathbb {R}$-Cartier, and 
$(X,\Delta )$ is klt as defined in [Reference Kollár and MoriKM98].
The following invariant was first introduced in [Reference LiLi18] and plays an important role in the study of K-semistable Fano varieties and Fano cone singularities.
Definition 2.8 For a klt singularity 
$x\in (X,\Delta )$, the normalized volume function 
$\widehat {\mathrm {vol}}_{(X,\Delta ),x}:\mathrm {Val}_{X,x}\to (0,+\infty ]$ is defined by
\[ \widehat{\mathrm{vol}}_{(X,\Delta),x}(v):= \left\{\begin{array}{ll} A_{X,\Delta}(v)^n\cdot \mathrm{vol}(v), & \text{if } A_{X,\Delta}(v)<+\infty,\\ +\infty, & \text{if } A_{X,\Delta}(v)=+\infty, \end{array} \right. \]
where 
$A_{X,\Delta }(v)$ is the log discrepancy of 
$v$ as defined in [Reference Jonsson and MustaţăJM12, Reference Boucksom, de Fernex, Favre and UrbinatiBdFFU15]. The local volume of a klt singularity 
$x\in (X,\Delta )$ is defined as
\[ \widehat{\mathrm{vol}}(x,X,\Delta):= \inf_{v\in\mathrm{Val}_{X,x}} \widehat{\mathrm{vol}}_{(X,\Delta),x}(v). \]
The above infimum is indeed a minimum by [Reference BlumBlu18, Reference XuXu20].
3. Saturation
Throughout this section, 
$(R,\mathfrak {m},\kappa )$ denotes an 
$n$-dimensional, analytically irreducible, Noetherian, local domain.
3.1 Definition of the saturation
Let 
$\mathfrak {a}\subset R$ be an 
$\mathfrak {m}$-primary ideal. Recall that the integral closure 
$\overline {\mathfrak {a}}$ of 
$\mathfrak {a}$ can be characterized valuatively by
\[ \overline{\mathfrak{a}}=\{f\in R\mid v(f)\ge v(\mathfrak{a}) \text{ for all } v\in\mathrm{DivVal}_{R,\mathfrak{m}}\}. \]
See, for example, [Reference Huneke and SwansonHS06, Theorem 6.8.3] and [Reference LazarsfeldLaz04, Example 9.6.8]. We define the saturation of an 
$\mathfrak {m}$-filtration in a similar manner.
Definition 3.1 The saturation 
$\widetilde {\mathfrak {a}}_\bullet$ of an 
$\mathfrak {m}$-filtration 
$\mathfrak {a}_\bullet$ is defined by
\[ \widetilde{\mathfrak{a}}_\lambda:= \{f\in \mathfrak{m} \mid v(f)\ge \lambda\cdot v(\mathfrak{a}_\bullet) \text{ for all } v\in\mathrm{DivVal}_{R,\mathfrak{m}} \} \]
for each 
$\lambda \in \mathbb {R}_{>0}$. We say 
$\mathfrak {a}_\bullet$ is saturated if 
$\mathfrak {a}_\bullet =\widetilde {\mathfrak {a}}_\bullet$.
Remark 3.2 Proposition 3.19 shows that it is equivalent to define the saturation using all positive volume valuations, rather than only divisorial valuations.
Remark 3.3 The saturation is a local analogue of the maximal norm of a multiplicative norm of the section ring of a polarized variety, which was defined and studied in [Reference Boucksom and JonssonBJ21]. See § 6.
Remark 3.4 Definition 3.1 differs from the definition of saturation used in [Reference MustaMus02, § 2] for monomial ideals, which coincides with the ideals in Lemma 3.6 defined using the integral closure of the Rees algebra.
Remark 3.5 After the first version of this paper was posted on the arXiv, Cutkosky and Praharaj introduced an operation on 
$\mathbb {R}$-filtrations that is defined using certain asymptotic Hilbert–Samuel functions (see the definition in [Reference Cutkosky and PraharajCP22, Theorem 1.3]). As shown by [Reference Cutkosky and PraharajCP22, Example 7.2], their operation does not always coincide with the saturation.
Lemma 3.6 [Reference CutkoskyCut21, Lemma 5.6]
If 
$\mathfrak {a}_\bullet$ be an 
$\mathfrak {m}$-filtration, then the integral closure of 
${\rm Rees}(\mathfrak {a}_\bullet ) := \bigoplus _{m \in \mathbb {N}} \mathfrak {a}_m t^m$ in 
$R[t]$ is given by 
$\bigoplus _{m\in \mathbb {N}} {\mathfrak {a}}'_m t^m$, where
\[ {\mathfrak{a}}'_m=\{f\in R \mid\text{there exists } r>0 \text{ such that } f^r\in \overline{\mathfrak{a}}_{rm}\}. \]
Example 3.7 In general, the ideals 
$\widetilde {\mathfrak {a}}_m$ and 
${\mathfrak {a}}'_m$ appearing above do not coincide. For example, let 
$R:= k[x]_{(x)}$ and 
$\mathfrak {m}=(x)$. Consider the 
$\mathfrak {m}$-filtration 
$\mathfrak {a}_\bullet$ defined by 
$\mathfrak {a}_\lambda := (x^{\lceil \lambda +1\rceil })$. Using that 
$\mathrm {ord}_{\mathfrak {m}} (\mathfrak {a}_\lambda ) =\lceil \lambda +1 \rceil$ and 
$\mathrm {ord}_{\mathfrak {m}} (\mathfrak {a}_\bullet )= 1$, we compute
\[ {\mathfrak{a}}'_m = (x^{m+1}) \subsetneq (x^m) = \widetilde{\mathfrak{a}}_m \]
for each 
$m\in \mathbb {Z}_{>0}$. In particular, 
${\rm Rees}(\mathfrak {a}_\bullet )$ is integrally closed, but 
$\mathfrak {a}_\bullet$ is not saturated.
The following lemma states basic properties of the saturation.
Lemma 3.8 For any 
$\mathfrak {m}$-filtration 
$\mathfrak {a}_\bullet$, the following statements hold:
(1)
$\mathfrak {a}_\bullet \subset \widetilde {\mathfrak {a}}_\bullet$;(2)
$v(\mathfrak {a}_\bullet )=v(\widetilde {\mathfrak {a}}_\bullet )$ for all $v\in \mathrm {DivVal}_{R,\mathfrak {m}}$; and(3)
$\widetilde {\mathfrak {a}}_\bullet$ is saturated.
Proof. For any 
$\lambda \in \mathbb {R}_{>0}$,
\[ v(\mathfrak{a}_\lambda) = \lim_{m \to \infty} \frac{ v(\mathfrak{a}_\lambda^m)}{m} \geq \lim_{m\to\infty} \biggl( \frac{ v\big(\mathfrak{a}_{\lfloor \lambda m\rfloor }\big)}{\lfloor \lambda m\rfloor } \cdot \frac{ \lfloor \lambda m \rfloor}{m} \biggr)= \lambda v(\mathfrak{a}_\bullet) , \]
where the inequality uses that 
$(\mathfrak {a}_{\lambda })^m \subset \mathfrak {a}_{\lambda m} \subset \mathfrak {a}_{ \lfloor \lambda m \rfloor }$. Therefore, 
$\mathfrak {a}_{\lambda } \subset \widetilde {\mathfrak {a}}_\lambda$, which is (1).
For (2), note that 
$v(\mathfrak {a}_\bullet ) \leq {v(\widetilde {\mathfrak {a}}_m ) }/{m} \leq {v(\mathfrak {a}_m)}/{m}$ for each 
$m \in \mathbb {Z}_{>0}$, where the first inequality follows from the definition of 
$\widetilde {\mathfrak {a}}_m$ and the second from (1). Sending 
$m\to \infty$ gives 
$v(\mathfrak {a}_\bullet ) \leq v(\widetilde {\mathfrak {a}}_\bullet )\leq v(\mathfrak {a}_\bullet )$, which implies (2). Statement (3) follows immediately from (2).
We say two 
$\mathfrak {m}$-filtrations 
$\mathfrak {a}_\bullet$ and 
$\mathfrak {b}_\bullet$ are equivalent if 
$\widetilde {\mathfrak {a}}_\bullet = \widetilde {\mathfrak {b}}_\bullet$. The following proposition gives a characterization of when two filtrations are equivalent after possible scaling.
Proposition 3.9 Let 
$\mathfrak {a}_\bullet$ and 
$\mathfrak {b}_\bullet$ be 
$\mathfrak {m}$-filtrations and 
$c\in \mathbb {R}_{>0}$. The following statements are equivalent.
(1)
$\widetilde {\mathfrak {a}}_\bullet = \widetilde {\mathfrak {b}}_{c \bullet }$.(2)
$v(\mathfrak {a}_\bullet ) = c v(\mathfrak {b}_\bullet )$ for all $v\in \mathrm {DivVal}_{R,\mathfrak {m}}$.
Proof. First, note that 
$v(\mathfrak {b}_{c \bullet }) =c v(\mathfrak {b}_\bullet )$ for all 
$v\in \mathrm {Val}_{R,\mathfrak {m}}$. Therefore, (1) implies (2) follows from Lemma 3.8(2), while (2) implies (1) follows from the definition of the saturation.
3.2 Saturation and completion
Let 
$(\widehat {R},\widehat {\mathfrak {m}})$ denote the 
$\mathfrak {m}$-adic completion of 
$(R,\mathfrak {m})$ and write 
$\varphi :R\to \hat {R}$ for the natural morphism. (Note that 
$\widehat {R}$ is a domain by the assumption throughout the paper that 
$R$ is analytically irreducible.) For an 
$\mathfrak {m}$-filtration 
$\mathfrak {a}_\bullet$, we set 
$\mathfrak {a}_\bullet \widehat {R} : = (\mathfrak {a}_\lambda \widehat {R})_{\lambda >0}$, which is an 
$\widehat {\mathfrak {m}}$-filtration.
Proposition 3.10 If 
$\mathfrak {a}_\bullet$ is an 
$\mathfrak {m}$-filtration and 
$\mathfrak {b}_\bullet :=\mathfrak {a}_\bullet \widehat {R}$, then 
$\widetilde {\mathfrak {a}}_\bullet \widehat {R} = \widetilde {\mathfrak {b}}_\bullet$.
The proposition shows that completion and saturation commute, as is the case with the integral closure of ideals [Reference Huneke and SwansonHS06, Proposition 1.6.2]. As a consequence of the proposition, many results regarding saturations reduce to the case when 
$R$ is a complete local domain.
Proof. By [Reference Huneke and SwansonHS06, Theorem 9.3.5], there is a bijective map 
$\varphi _* \colon \mathrm {DivVal}_{\widehat {R},\widehat {\mathfrak {m}}} \to \mathrm {DivVal}_{R,\mathfrak {m}}$ that sends 
$\hat {v}$ to the valuation 
$v$ defined by composition
\[ {\rm Frac}(R)^\times \hookrightarrow {\rm Frac}(\hat{R})^\times \overset{\hat{v}}{\to} \mathbb{R} . \]
Note that 
$\hat {v}(\mathfrak {a} \hat {R}) = v(\mathfrak {a})$ for any ideal 
$\mathfrak {a}\subset R$.
Fix 
$\lambda \in \mathbb {R}_{>0}$. Using the previous observation twice and the definition of the saturation, we compute
\[ \hat{v} (\widetilde{\mathfrak{a}}_\lambda \hat{R} ) = v(\widetilde{\mathfrak{a}}_\lambda) \geq \lambda v (\mathfrak{a}_\bullet) = \lambda \hat{v}(\mathfrak{b}_\bullet) . \]
Thus, 
$\widetilde {\mathfrak {a}}_{\lambda } \hat {R} \subset \widetilde {\mathfrak {b}}_\lambda$. To prove the reverse inclusion, note that 
$\mathfrak {c} R = \widetilde {\mathfrak {b}}_\lambda$ for some ideal 
$\mathfrak {c}\subset R$, since 
$\widetilde {\mathfrak {b}}_\lambda$ is 
$\widehat {\mathfrak {m}}$-primary. As before, we compute
\[ v(\mathfrak{c})= \hat{v}(\widetilde{\mathfrak{b}}_\lambda) \geq \lambda \hat{v}(\mathfrak{b}_\bullet) =\lambda v (\mathfrak{a}_\bullet) . \]
Therefore, 
$\widetilde {\mathfrak {b}}_\lambda = \mathfrak {c} \hat {R} \subset \widetilde {\mathfrak {a}}_\lambda \hat {R}$.
Proposition 3.11 If 
$\mathfrak {a}_\bullet$ is an 
$\mathfrak {m}$-filtration and 
$\mathfrak {b}_\bullet = \mathfrak {a}_\bullet \widehat {R}$, then 
$\mathrm {e}(\mathfrak {a}_\bullet ) = \mathrm {e}(\mathfrak {b}_\bullet )$.
Proof. Since 
$R/\mathfrak {a}_\lambda$ and 
$\widehat {R}/\mathfrak {b}_\lambda$ are isomorphic as Artinian rings, 
$\ell (R/\mathfrak {a}_\lambda )=\ell (\widehat {R}/\mathfrak {b}_\lambda )$ for each 
$\lambda >0$. Therefore, the equality of multiplicities holds.
3.3 Saturation and multiplicity
In this section we prove Theorem 1.4 and a number of corollaries. The theorem is a consequence of the following two propositions.
Proposition 3.12 Let 
$\mathfrak {a}_\bullet \subset \mathfrak {b}_\bullet$ be 
$\mathfrak {m}$-filtrations. If 
$\mathrm {e}(\mathfrak {a}_\bullet )= \mathrm {e}(\mathfrak {b}_\bullet )$, then 
$v(\mathfrak {a}_\bullet ) = v(\mathfrak {b}_\bullet )$ for all 
$v\in \mathrm {Val}_{R,\mathfrak {m}}^+$.
The proposition was shown when 
$v$ is divisorial in [Reference CutkoskyCut21, Theorem 7.3] using Okounkov bodies. The proof below instead follows the strategy of [Reference Li and XuLX20, Proposition 2.7] and [Reference Xu and ZhuangXZ21, Lemma 3.9].
Proof. Suppose the statement is false. Then there exists 
$v\in \mathrm {Val}_{R,\mathfrak {m}}^+$ such that 
$v(\mathfrak {b}_\bullet ) < v(\mathfrak {a}_\bullet )$. After scaling 
$v$, we may assume 
$v(\mathfrak {a}_\bullet )=1$. Therefore, there exists 
$l \in \mathbb {Z}_{>0}$ so that 
$v(\mathfrak {b}_{l}) /l < v(\mathfrak {a}_\bullet )=1$. Hence, we may choose 
$f\in \mathfrak {b}_l\setminus \mathfrak {a}_l$ such that 
$k:= v(f)< l$.
Now, consider the map
\[ \phi: R \to \mathfrak{b}_{l m}/ \mathfrak{a}_{lm}\quad \text{sending}\quad g \mapsto f^m g. \]
We claim 
$\ker (\phi ) \subset \mathfrak {a}_{ (l-k)m } (v)$. Indeed, if 
$g\in \ker \phi$, then 
$f^m g \in \mathfrak {a}_{lm}$. Hence,
\[ v(g) =v(f^m g) - v(f^m) \geq v(\mathfrak{a}_{l m}) - km \geq l m - km = (l-k) m \]
as desired. Using the claim, we deduce
\begin{equation} \ell( \mathfrak{b}_{l m}/ \mathfrak{a}_{lm}) \geq \ell (R/ \ker \phi) \geq \ell ( R/ \mathfrak{a}_{ (l-k)m } (v)) . \end{equation}Finally, we compute
\[ \mathrm{e}(\mathfrak{a}_\bullet) - \mathrm{e}(\mathfrak{b}_\bullet) = \lim_{ m \to \infty} \frac{ \ell (\mathfrak{b}_{lm } / \mathfrak{a}_{lm})}{(lm)^n /n!} \geq \lim_{m\to\infty} \frac{\ell ( R/ \mathfrak{a}_{ (l-k) m }(v) )}{ (lm)^n /n!} = \frac{(l-k)^n \,\mathrm{vol}(v)}{l^n} >0 , \]
where the first inequality is by (3.1) and the second uses that 
$v$ has positive volume. This contradicts our assumption that 
$\mathrm {e}(\mathfrak {a}_\bullet )=\mathrm {e}(\mathfrak {b}_\bullet )$.
Proposition 3.13 Let 
$\mathfrak {a}_\bullet$ and 
$\mathfrak {b}_\bullet$ be 
$\mathfrak {m}$-filtrations. Assume 
$(R,\mathfrak {m})$ is complete. If 
$v(\mathfrak {b}_\bullet ) \leq v(\mathfrak {a}_\bullet )$ for all 
$v\in \mathrm {DivVal}_{R,\mathfrak {m}}$, then 
$\mathrm {e}(\mathfrak {b}_\bullet ) \leq \mathrm {e}(\mathfrak {a}_\bullet )$.
The proposition and its proof are local analogues of [Reference SzékelyhidiSzé15, Lemma 22], which concerns the volumes of graded linear series of projective varieties.
Remark 3.14 The assumption that 
$R$ is complete in Proposition 3.13 implies that 
$R$ is Nagata [Sta, Lemma 032W] (see also [Sta, Definition 033S]). The latter property will be used in the proof of Proposition 3.13 to ensure certain normalization morphisms are proper.
Proof. We claim that there exists a sequence of proper birational morphisms
\[ \cdots\to X_3 \to X_2\to X_1\to X_0 = X \]
such that the each 
$X_i$ is normal and the sheafs 
$\mathfrak {a}_{l} \cdot \mathcal {O}_{X_m}$ and 
$\mathfrak {b}_{l}\cdot \mathcal {O}_{X_m}$ are line bundles when 
$l\leq m$. Such a sequence can be constructed inductively as follows. Let 
$X_{i} \to X_{i-1}$ be defined by the composition
\[ X_{i} \to X_{i,2}\to X_{i,1} \to X_{i-1}, \]
where 
$X_{i,1}$ is the blowup of 
$\mathfrak {a}_i \cdot \mathcal {O}_{X_{i-1}}$, 
$X_{i,2}$ is the blowup of 
$\mathfrak {b}_i \cdot \mathcal {O}_{X_{i,1}}$, and 
$X_i$ the normalization of 
$X_{i,2}$. Note that 
$X_{i,2} \to X$ is proper, since 
$X_{i,2}\to X_{i-1}$ is a blowup and 
$X_{i-1}\to X$ is proper by our inductive assumption. Since 
$X$ is Nagata and 
$X_{i,2}\to X$ is finite type, 
$X_{i,2}$ is Nagata by [Sta, Lemma 035A]. Therefore, 
$X_i\to X_{i,2}$ is finite by the definition of Nagata, and we conclude the composition 
$X_i \to X_{i-1}$ is proper, which completes the proof of the claim.
Next, consider a proper birational morphism 
$Y\to X$ with 
$Y$ normal and factoring as
\[ Y \overset{\pi_m}{\longrightarrow} X_m \longrightarrow X, \]
For 
$l\leq m$, there exist Cartier divisors 
$G_l$ and 
$G'_l$ on 
$Y$ such that
\[ \mathfrak{a}_l \cdot \mathcal{O}_{Y} = \mathcal{O}_{Y} (G_l) \quad \text{ and } \quad \mathfrak{b}_l\cdot\mathcal{O}_{Y}=\mathcal{O}_{Y}(G_l') . \]
Set 
$F_l:= {G_l}/{l}$ and 
$F'_l:= {G'_l}/{l}$, which are relatively nef over 
$X$ and satisfy
\[ F_l=- \sum_{E \subset Y} \frac{\mathrm{ord}_E(\mathfrak{a}_l)}{l} E \quad \text{ and }\quad F'_l =- \sum_{E \subset Y} \frac{\mathrm{ord}_E(\mathfrak{b}_l)}{l} E, \]
where the sums run through prime divisors 
$E\subset Y$. Throughout the proof, we will without mention replace 
$Y$ with higher birational models so it factors through certain 
$X_m\to X$.
Given 
$\epsilon >0$, by Proposition 2.6 and (2.1), there exists 
$m_0>0$ such that
\begin{equation} - F_{m_0}^{n-1} \cdot F_{m_0} =\mathrm{e}(\mathfrak{a}_{m_0})/{m_0}^n< \mathrm{e}(\mathfrak{a}_\bullet) + \epsilon. \end{equation}
For a multiple 
$m_1$ of 
$m_0$,
\[ -(F_{m_0})^{n-1} \cdot F_{m_1}< \mathrm{e}(\mathfrak{a}_\bullet) + \epsilon , \]
since 
$F_{m_0} \leq F_{m_1}$ and 
$F_{m_0}$ is nef over 
$X$. Now, we compute
\begin{align*} - (F_{m_0})^{n-1} \cdot
F_{m_1} + (F_{m_0})^{n-1} \cdot F'_{m_1} &= \sum_{E\subset
Y } \big(\!- \mathrm{ord}_{E} (F_{m_1}) +\mathrm{ord}_{E}
(F'_{m_1}) \big) (F_{m_0})^{n-1} \cdot E\\ &=
\sum_{E\subset X_{m_0} } \big(\!-\mathrm{ord}_{E} (F_{m_1})
+\mathrm{ord}_{E} (F'_{m_1}) \big) ({\pi_{m_0}}_*
F_{m_0})^{n-1} \cdot E, \end{align*}
where the second equality uses that 
$F_{m_0}$ is the pullback of 
${\pi _{m_0}}_* F_{m_0}$ and the projection formula [Reference KleimanKle05, Proposition B.16]. Since 
${\pi _{m_0}}_* F_{m_0}$ is nef over 
$X$ and
\[ \lim_{m\to \infty} \big(\!-\mathrm{ord}_{E} (F_m) + \mathrm{ord}_{E}(F'_m) \big) = \mathrm{ord}_E(\mathfrak{a}_\bullet) -\mathrm{ord}_E(\mathfrak{b}_\bullet) \geq 0 , \]
we may choose 
$m_1$ sufficiently large and divisible by 
$m_0$ such that
\[ -(F_{m_0})^{n-1} \cdot F'_{m_1} \leq - F_{m_0}^{n-1} \cdot F_{m_1} + \epsilon. \]
Therefore,
\[ - (F_{m_0})^{n-2} \cdot F'_{m_1} \cdot F_{m_0} = - (F_{m_0})^{n-1} \cdot F'_{m_1} \leq \mathrm{e}(\mathfrak{a}_\bullet) + 2\epsilon , \]
where the first equality is Proposition 2.5.
For any multiple 
$m_2$ of 
$m_1$,
\[ - (F_{m_0})^{n-2} \cdot F'_{m_1} \cdot F_{m_2} \leq \mathrm{e}(\mathfrak{a}_\bullet) + 2\epsilon, \]
since 
$F_{m_0} \leq F_{m_2}$ and the terms 
$F'_{m_1}$ and 
$F_{m_0}$ are nef over 
$X$. Similarly to the previous paragraph, we compute
\begin{align*} &- (F_{m_0})^{n-2} \cdot F'_{m_1} \cdot F_{m_2} - (F_{m_0})^{n-2} \cdot F'_{m_1} \cdot F'_{m_2}\\ &\qquad =\sum_{E \subset X_{m_1} } \big(\!- \mathrm{ord}_{E} (F_{m_2})+ \mathrm{ord}_{E} (F'_{m_2}) \big) ({ ({\pi_{m_1}}_* F_{m_0})^{n-1} \cdot \pi_{m_1}}_*F'_{m_1}) \cdot E. \end{align*}
and, hence, we may choose 
$m_2$ sufficiently large and divisible by 
$m_1$ so that
\[ -(F_{m_0})^{n-2} \cdot F'_{m_1} \cdot F'_{m_2} <-(F_{m_0})^{n-2} \cdot F'_{m_1} \cdot F_{m_2}+ \epsilon. \]
Thus,
\[ -(F_{m_0})^{n-2} \cdot F'_{m_1} \cdot F'_{m_2} < \mathrm{e}(\mathfrak{a}_\bullet) +3\epsilon. \]
Repeating in this way gives
\[ -F'_{m_1} \cdot F'_{m_2} \cdot \cdots \cdot F'_{m_n} < \mathrm{e}(\mathfrak{a}_\bullet)+ (n+1)\epsilon , \]
where each 
$m_i$ divides 
$m_{i+1}$. Since each 
$F'_{m_i}$ is nef over 
$X$ and 
$F'_{m_{i}} \leq F'_{m_{i+1}}$, we see that
\[ -(F'_{m_n})^n < \mathrm{e}(\mathfrak{a}_\bullet)+ n\epsilon. \]
Using that 
$\mathrm {e}(\mathfrak {b}_\bullet ) = \inf _{m} \mathrm {e}(\mathfrak {b}_m)/m^n =\inf _{m} -(F'_{m})^n$ by Lemma 2.6 and (2.1), we see that 
$\mathrm {e}(\mathfrak {b}_\bullet ) < \mathrm {e}(\mathfrak {a}_\bullet ) + (n+1)\epsilon$. Therefore, 
$\mathrm {e}(\mathfrak {b}_\bullet )\leq \mathrm {e}(\mathfrak {a}_\bullet )$.
Remark 3.15 When 
$x\in X$ is an isolated singularity on a normal variety, Proposition 3.13 follows easily from the intersection theory for nef 
$b$-divisors developed in [Reference Boucksom, de Fernex and FavreBdFF12]. Indeed, the assumption 
$v(\mathfrak {a}_\bullet ) \leq v(\mathfrak {b}_\bullet )$ implies 
$Z(\mathfrak {a}_\bullet )\ge Z(\mathfrak {b}_\bullet )$, where 
$Z(\mathfrak {a}_\bullet )$ is the nef 
$b$-Weil 
$\mathbb {R}$-divisor associated to 
$\mathfrak {a}_\bullet$. The proposition then follows from [Reference Boucksom, de Fernex and FavreBdFF12, Remark 4.17]. However, when 
$x$ is not an isolated singularity, a satisfactory intersection theory for nef 
$b$-divisors seems missing from the literature.
We are now ready to prove Theorem 1.4 and a number of its corollaries.
Proof of Theorem 1.4 By Propositions 3.10 and 3.11, we may assume 
$(R,\mathfrak {m})$ is complete. This condition will be needed below to apply Proposition 3.13.
By Propositions 3.12 and 3.13, 
$\mathrm {e}(\mathfrak {a}_\bullet ) = \mathrm {e}(\mathfrak {b}_\bullet )$ if and only if 
$v(\mathfrak {a}_\bullet ) = v(\mathfrak {b}_\bullet )$ for all 
$v\in \mathrm {DivVal}_{R,\mathfrak {m}}$. By Proposition 3.9, the latter condition holds if and only if 
$\widetilde {\mathfrak {a}}_\bullet = \widetilde {\mathfrak {b}}_\bullet$.
Corollary 3.16 If 
$\mathfrak {a}_\bullet$ is an 
$\mathfrak {m}$-filtration, then 
$\mathrm {e}(\mathfrak {a}_\bullet )=\mathrm {e}(\widetilde {\mathfrak {a}}_\bullet )$.
Proof. By Lemma 3.8, 
$\mathfrak {a}_\bullet \subset \widetilde {\mathfrak {a}}_\bullet$ and 
$\widetilde {\mathfrak {a}}$ is saturated. Thus, Theorem 1.4 implies 
$\mathrm {e}(\mathfrak {a}_\bullet ) = \mathrm {e}(\widetilde {\mathfrak {a}}_\bullet )$.
Corollary 3.17 Let 
$\mathfrak {a}_\bullet$ and 
$\mathfrak {b}_\bullet$ be 
$\mathfrak {m}$-filtrations. The following statements are equivalent.
(1)
$\widetilde {\mathfrak {a}}_\bullet = \widetilde {\mathfrak {b}}_{ \bullet }$.(2)
$\mathrm {e}(\mathfrak {a}_\bullet ) = \mathrm {e}(\mathfrak {a}_\bullet \cap \mathfrak {b}_{\bullet })= \mathrm {e}(\mathfrak {b}_\bullet )$.
Proof of Corollary 3.17 Assume (1) holds. Observe that
\[ \mathrm{e}(\mathfrak{a}_\bullet)=\mathrm{e}(\widetilde{\mathfrak{a}}_\bullet) = \mathrm{e}(\widetilde{\mathfrak{b}}_{\bullet}) = \mathrm{e}(\mathfrak{b}_\bullet) , \]
where the first and third equality follow from Corollary 3.16. Thus, it remains to show that 
$\mathrm {e}(\mathfrak {a}_\bullet \cap \mathfrak {b}_\bullet )= \mathrm {e}(\mathfrak {b}_\bullet )$. First, note that 
$\mathrm {e}(\mathfrak {a}_\bullet \cap \mathfrak {b}_\bullet ) \geq \mathrm {e}(\mathfrak {b}_\bullet )$ holds trivially, since 
$\mathfrak {a}_\bullet \cap \mathfrak {b}_\bullet \subset \mathfrak {b}_\bullet$. For the reverse inequality, we compute
\begin{align*} \mathrm{e}(\mathfrak{a}_\bullet\cap \mathfrak{b}_{\bullet}) = \lim_{m \to \infty} \frac{ \ell (R/(\mathfrak{a}_m \cap \mathfrak{b}_{ m}))}{m^n/n!} &= \lim_{m \to \infty} \biggl( \frac{ \ell (R/(\mathfrak{a}_m) )}{m^n/n!} + \frac{\ell(R/\mathfrak{b}_{ m}) }{m^n/n!}-\frac{ \ell(R/(\mathfrak{a}_m+\mathfrak{b}_{ m}))}{m^n/n!}\biggr)\\ &= \mathrm{e}(\mathfrak{a}_\bullet) +\mathrm{e}(\mathfrak{b}_{ \bullet}) - \lim_{m\to \infty} \frac{ \ell(R/(\mathfrak{a}_m+\mathfrak{b}_{ m}))}{m^n/n!}\\ &\leq \mathrm{e}(\mathfrak{a}_\bullet) +\mathrm{e}(\mathfrak{b}_{ \bullet}) - \lim_{m\to \infty} \frac{ \ell(R/\widetilde{\mathfrak{a}}_m)}{m^n/n!}\\ &= \mathrm{e}(\mathfrak{a}_\bullet) +\mathrm{e}(\mathfrak{b}_{ \bullet}) - \mathrm{e}(\widetilde{\mathfrak{a}}_\bullet)\\ & = \mathrm{e}(\mathfrak{b}_\bullet), \end{align*}
where the inequality uses that 
$\mathfrak {a}_m+\mathfrak {b}_m \subset \widetilde {\mathfrak {a}}_m + \widetilde {\mathfrak {b}}_m = \widetilde {\mathfrak {a}}_m+\widetilde {\mathfrak {a}}_m = \widetilde {\mathfrak {a}}_m$ by the assumption that (1) holds. Therefore, (2) holds.
Conversely, assume (2) holds. Applying Theorem 1.4 to both 
$(\mathfrak {a}_\bullet \cap \mathfrak {b}_\bullet ) \subset \mathfrak {a}_\bullet$ and 
$(\mathfrak {a}_\bullet \cap \mathfrak {b}_\bullet )\subset \mathfrak {b}_\bullet$, we see that 
$\widetilde {\mathfrak {a}_\bullet \cap \mathfrak {b}_{ \bullet }}= \widetilde {\mathfrak {a}}_\bullet$ and 
$\widetilde {\mathfrak {a}_\bullet \cap \mathfrak {b}_{ \bullet }}= \widetilde {\mathfrak {b}}_\bullet$, which implies that (1) holds.
The following result was proven when 
$R$ is regular in [Reference MustaMus02, Theorem 1.7.2].
Corollary 3.18 An 
$\mathfrak {m}$-filtration 
$\mathfrak {a}_\bullet$ is linearly bounded if and only if 
$\mathrm {e}(\mathfrak {a}_\bullet )>0$.
Proof. If 
$\mathfrak {a}_\bullet$ is linearly bounded, then there exists 
$c>0$ such that 
$\mathfrak {a}_{\lambda } \subset \mathfrak {m}^{\lceil c \lambda \rceil }$ for all 
$\lambda >0$. Thus, 
$\mathrm {e}(\mathfrak {a}_\bullet ) \geq c^{n} \mathrm {e}(\mathfrak {m}) >0$ as desired.
Next, assume 
$\mathrm {e}(\mathfrak {a}_\bullet )>0$. We claim that there exists 
$v\in \mathrm {DivVal}_{R,\mathfrak {m}}$ such that 
$v(\mathfrak {a}_\bullet )>0$. If not, then 
$\widetilde {\mathfrak {a}}_\lambda = \mathfrak {m}$ for all 
$\lambda >0$. Using Corollary 3.16, we then see 
$\mathrm {e}(\mathfrak {a}_\bullet ) = \mathrm {e}(\widetilde {\mathfrak {a}}_\bullet ) =0$, which is a contradiction. Now, fix 
$v\in \mathrm {DivVal}_{R,\mathfrak {m}}$ with 
$v(\mathfrak {a}_\bullet ) >0$. Using that 
$\mathfrak {a}_\bullet \subset \mathfrak {a}_{\bullet }(v)$ and Proposition 2.7, we conclude 
$\mathrm {e}(\mathfrak {a}_\bullet )\geq \mathrm {e}(\mathfrak {a}_\bullet (v)) >0$.
3.4 Saturation and finite-volume valuations
Using results from the previous section, we show that the saturation can be defined using positive volume valuations, rather than only divisorial valuations.
Proposition 3.19 If 
$\mathfrak {a}_\bullet$ is an 
$\mathfrak {m}$-filtration and 
$\lambda \in \mathbb {R}_{>0}$, then
\[ \widetilde{\mathfrak{a}}_\lambda = \big\{ f\in \mathfrak{m} \mid v(f) \geq \lambda \cdot v(\mathfrak{a}_\bullet) \text{ for all } v\in \mathrm{Val}_{R,\mathfrak{m}}^+ \big\}. \]
The proposition will be deduced from the following lemma.
Lemma 3.20 If 
$\{ v_i \}_{i\in I}$ is a collection of valuations in 
$\mathrm {Val}_{R,\mathfrak {m}}^+$ and 
$\{ c_i \}_{i\in I}$ of non-negative real numbers, then the 
$\mathfrak {m}$-filtration 
$\mathfrak {a}_\bullet$ defined by
\[ \mathfrak{a}_\lambda := \{ f\in \mathfrak{m} \mid v_i (f) \geq \lambda c_i \text{ for all } i \in I\} \]
is saturated. Hence, 
$\mathfrak {a}_\bullet (v)$ is saturated for any 
$v\in \mathrm {Val}_{R,\mathfrak {m}}^+$.
Proof. Suppose the statement is false. Then there exists some 
$\lambda \in \mathbb {R}_{>0}$ such that 
$\mathfrak {a}_\lambda \subsetneq \widetilde {\mathfrak {a}}_{\lambda }$. Thus, there exist 
$f\in \widetilde {\mathfrak {a}}_{\lambda }$ and 
$i \in I$ such that 
$v_i(f) < \lambda c_i$. We claim 
$v_i(\widetilde {\mathfrak {a}}_\bullet )< v_i(\mathfrak {a}_\bullet )$. Indeed,
\[ v_i(\widetilde{\mathfrak{a}}_\bullet)= \lim_{m \to \infty} \frac{ v_i(\widetilde{\mathfrak{a}}_{ \lfloor \lambda m\rfloor })}{ \lfloor \lambda m \rfloor } \leq \lim_{m\to \infty} \frac{ v_i(f^m)}{ \lfloor \lambda m \rfloor } = \frac{ v_i(f) }{\lambda}< c_i, \]
where the second equality uses that 
$f^m \in (\widetilde {\mathfrak {a}}_{\lambda })^m \subset \widetilde {\mathfrak {a}}_{\lambda m} \subset \widetilde {\mathfrak {a}}_{\lfloor \lambda m \rfloor }$. Since 
$v_i(\mathfrak {a}_\bullet ) \geq c_i$, the claim holds.
By Proposition 3.12 and the claim, 
$\mathrm {e}(\widetilde {\mathfrak {a}}_\bullet )< \mathrm {e}(\mathfrak {a}_\bullet )$. The latter contradicts Theorem 1.4.
Proof of Proposition 3.19 Let 
$\mathfrak {b}_\bullet$ denote the 
$\mathfrak {m}$-filtration defined by
\[ \mathfrak{b}_\lambda := \{f \in \mathbb{R} \mid v(f) \geq \lambda \cdot v(\mathfrak{a}_\bullet) \text{ for all } v \in \mathrm{Val}_{R,\mathfrak{m}}^+ \} . \]
Notice that 
$\mathfrak {a}_\bullet \subset \mathfrak {b}_\bullet \subset \widetilde {\mathfrak {a}}_\bullet$, where the second inclusion uses that all divisorial valuations have positive volume. Taking saturations gives 
$\widetilde {\mathfrak {a}}_\bullet \subset \widetilde {\mathfrak {b}}_\bullet \subset \widetilde {\widetilde {\mathfrak {a}}}_\bullet$. Since 
$\mathfrak {b}_\bullet$ and 
$\widetilde {\mathfrak {a}}_\bullet$ are saturated by Lemma 3.20, we conclude 
$\widetilde {\mathfrak {a}}_\bullet \subset \mathfrak {b}_\bullet \subset {\widetilde {\mathfrak {a}}}_\bullet$.
4. Multiplicity and geodesics
In this section, we prove Theorem 1.1 on the convexity of the multiplicity function along geodesics. Throughout, 
$(R,\mathfrak {m},\kappa )$ denotes an 
$n$-dimensional, analytically irreducible, Noetherian local domain containing a field of arbitrary characteristic.
4.1 Geodesics
Fix two finite-multiplicity 
$\mathfrak {m}$-filtrations 
$\mathfrak {a}_{\bullet,0}$ and 
$\mathfrak {a}_{\bullet,1}$.
Definition 4.1 For each 
$t\in (0,1)$, we define an 
$\mathfrak {m}$-filtration 
$\mathfrak {a}_{\bullet,t}$ by setting
\[ \mathfrak{a}_{\lambda,t} = \sum_{\lambda=(1-t)\mu+t\nu} \mathfrak{a}_{\mu,0} \cap \mathfrak{a}_{\nu,1} , \]
where the sum runs through all 
$\mu, \nu \in \mathbb {R}$ satisfying 
$\lambda =(1-t)\mu +t\nu$. We call 
$(\mathfrak {a}_{\bullet,t})_{t\in [0,1]}$ the geodesic between 
$\mathfrak {a}_{\bullet,0}$ and 
$\mathfrak {a}_{\bullet,1}$.
This definition is a local analog of the geodesic between two filtrations of the section ring of a polarized variety [Reference Blum, Liu, Xu and ZhuangBLXZ23, Reference RebouletReb22]. See Section 6 for details.
Example 4.2 Let 
$R:= k[x,y]_{(x,y)}$. For 
$\boldsymbol {\alpha }=(\boldsymbol {\alpha }_1,\boldsymbol {\alpha }_2) \in \mathbb {R}_{>0}^2$, let 
$v_{\boldsymbol {\alpha }}\colon {\rm Frac}(R)^\times \to \mathbb {R}$ be the monomial valuation with weight 
$\boldsymbol {\alpha }_1$ and 
$\boldsymbol {\alpha }_2$ with respect to 
$x$ and 
$y$, that is,
\[ v\biggl( \sum_{m,n \geq 0 } c_{m,n} x^{m}y^{n} \biggr) = \min \{ \boldsymbol {\alpha}_1m + \boldsymbol {\alpha}_2 n \mid c_{m,n} \neq 0\}. \]
Note that 
$\mathfrak {a}_\lambda (v_{\boldsymbol {\alpha }}) = \{x^{m}y^n \mid m \alpha _1 + n \alpha _2 \geq \lambda \}$. Now, fix 
$\boldsymbol {\alpha }, \boldsymbol {\beta } \in \mathbb {R}_{\geq 0}^2$ and consider the geodesic 
$\mathfrak {a}_{\bullet,t}$ between 
$\mathfrak {a}_{\bullet,0} := \mathfrak {a}_{\bullet }(v_{\boldsymbol {\alpha }})$ and 
$\mathfrak {a}_{\bullet,1}:= \mathfrak {a}_{\bullet }(v_{\boldsymbol {\beta }})$. A short computation shows
\[ \mathfrak{a}_{\bullet,t} = \mathfrak{a}_\bullet( v_{ (1-t)\boldsymbol {\alpha} + t \boldsymbol {\beta}}) . \]
This computation unfortunately does not generalize to the case of quasi-monomial valuations. See [Reference Liu, Xu and ZhuangLXZ22, § 4] for a study of this failure in the global setting.
To prove Theorem 1.1, we define a measure on 
$\mathbb {R}^2$ that encodes the multiplicities along the geodesic. The argument may be viewed as a local analogue of the construction in [Reference Blum, Liu, Xu and ZhuangBLXZ23, § 3.1] for the geodesic between two filtrations of the section ring of a polarized variety. The latter global construction is motivated by [Reference Boucksom and ChenBC11] and [Reference Boucksom, Hisamoto and JonssonBHJ17, Theorem 4.3], which constructs a measure on 
$\mathbb {R}$ associated to a single filtration of the section ring of polarized variety.
Before proceeding with the proof, fix integers 
$C>0$ and 
$D>0$ such that
\begin{gather*} \mathfrak{m}^{ \lceil C \lambda \rceil } \subset \mathfrak{a}_{\lambda,0} \cap \mathfrak{a}_{\lambda,1},\\ \mathfrak{a}_{D \lambda,0 } \subset \mathfrak{a}_{\lambda, 1} \quad\text{and}\quad \mathfrak{a}_{D\lambda,1} \subset \mathfrak{a}_{\lambda,0}. \end{gather*}
for all 
$\lambda \geq 1$. The existence of 
$C$ follows from Lemma 2.3, while the existence of 
$D$ follows from Lemma 2.3 and the fact that 
$\mathfrak {a}_{\bullet,0}$ and 
$\mathfrak {a}_{\bullet,1}$ are both linearly bounded by Corollary 3.18.
4.2 Sequences of measures
For each 
$m\in \mathbb {Z}_{>0}$, consider the function 
$H_m\colon \mathbb {R}^2 \to \mathbb {R}$ defined by
\[ H_m(x,y) := \frac{\ell \big( R/ (\mathfrak{a}_{mx,0} \cap \mathfrak{a}_{my,1}) \big)} { m^n/n! } . \]
Notice that 
$H_m$ is non-decreasing and left continuous in each variable. Using the sequence 
$(H_m)_{m\geq 1}$, we define a sequence of measures on 
$\mathbb {R}^2$.
Proposition 4.3 The distributional derivative 
$\mu _m:= - ({ \partial ^2 H_m}/{\partial x \partial y})$ is a discrete measure on 
$\mathbb {R}^2$ and has support contained in 
$\{ ({1}/{D}) x \leq y \leq Dx \} \cup \{ 0 \leq x \leq {1}/{m} \} \cup \{ 0 \leq y\leq {1}/{m} \}$.
Before proving the proposition, we prove the following lemma allowing us to reduce to the case of complete local rings.
Lemma 4.4 Let 
$(\widehat {R},\widehat {\mathfrak {m}})$ denote the completion of 
$(R,\mathfrak {m})$, 
$\mathfrak {b}_{\bullet,i}= \mathfrak {a}_{\bullet,i} \cdot \widehat {R}$ for 
$i=0,1$, and 
$(\mathfrak {b}_{t,\bullet })_{t\in [0,1]}$ the geodesic between 
$\mathfrak {b}_{\bullet,0}$ and 
$\mathfrak {b}_{\bullet,1}$. Then the following statements hold,
(1)
$\ell ( {R}/ \mathfrak {a}_{a, 0} \cap \mathfrak {a}_{b,1})= \ell ( \hat {R}/ \mathfrak {b}_{a, 0} \cap \mathfrak {b}_{b,1})$ for each $a,b\geq 0$.(2)
$\ell ( {R}/ \mathfrak {a}_{a,t}) = \ell ( \hat {R}/ \mathfrak {b}_{a, t} )$ for each $a\geq 0$ and $t\in [0,1]$.
Proof. For 
$\mathfrak {m}$-primary ideals 
$\mathfrak {c}\subset R$ and 
$\mathfrak {d}\subset R$,
\[ (\mathfrak{c} + \mathfrak{d} )\widehat{R} = \mathfrak{c} \widehat{R} + \mathfrak{d} \widehat{R} \quad \text{ and } \quad (\mathfrak{c} \cap \mathfrak{d} )\widehat{R} = \mathfrak{c} \widehat{R} \cap \mathfrak{d} \widehat{R} . \]
Therefore,
\[ (\mathfrak{a}_{a, 0} \cap \mathfrak{a}_{b,1} ) \widehat{R} = \mathfrak{b}_{a, 0} \cap \mathfrak{b}_{b,1} \quad \text{ and } \quad \mathfrak{a}_{a,t} \widehat{R} = \mathfrak{b}_{a, t} . \]
Statements (1) and (2) follow from these equalities.
Proof of Proposition 4.3 By Lemma 4.4, it suffices to prove the statement when 
$(R,\mathfrak {m})$ is complete. Since 
$R$ contains a field by assumption and is complete, there is an inclusion 
$\kappa \hookrightarrow R$ such that the composition 
$\kappa \hookrightarrow R \to \kappa$ is the identity. This will be helpful, since any finite-length 
$R$-module is naturally a finite-dimensional 
$\kappa$-vector space via restriction of scalars.
Fix an integer 
$N>0$ and consider the finite-dimensional 
$\kappa$-vector space 
$V:= R/ \mathfrak {m}^{NCm}$. The 
$\mathfrak {m}$-filtrations 
$\mathfrak {a}_{\bullet,0}$ and 
$\mathfrak {a}_{\bullet,1}$ induce decreasing filtrations 
$\mathcal {F}^\bullet _0$ and 
$\mathcal {F}^\bullet _1$ of 
$V$ defined by
\[ \mathcal{F}_0^\lambda V := {\rm im}( \mathfrak{a}_{\lambda,0} \to V) \quad \text{ and } \quad \mathcal{F}_1^\lambda V := {\rm im}(\mathfrak{a}_{\lambda,1} \to V) . \]
Observe that when 
$x< N$ and 
$y< N$,
\[ \ell \big( R / (\mathfrak{a}_{mx,0} \cap \mathfrak{a}_{my,1}) \big) = \dim_\kappa \big( V / (\mathcal{F}_0^{mx,0} V \cap \mathcal{F}_1^{my,1} V )\big) = \dim_\kappa V- \dim_\kappa (\mathcal{F}_0^{mx,0} V \cap \mathcal{F}_1^{my,1} V), \]
where the first equality uses that 
$\mathfrak {m}^{NCm} \subset \mathfrak {a}_{mx,0} \cap \mathfrak {a}_{my,1}$.
To analyze the dimension of 
$\mathcal {F}_0^{mx} V \cap \mathcal {F}_1^{my} V$, we use that any finite-dimensional vector space with two filtrations admits a basis simultaneously diagonalizing both filtrations (see [Reference Boucksom and ErikssonBE21, Proposition 1.14] and [Reference Abban and ZhuangAZ22, Lemma 3.1]). This means there exists a basis 
$(s_1,\ldots, s_\ell )$ for 
$V$ such that each 
$\mathcal {F}_j^\lambda V$ is the span of some subset of the basis elements. Hence, if we set 
$\lambda _{i,j} := \sup \{ \lambda \in \mathbb {R} \mid s_i \in \mathcal {F}_j^\lambda V \}$, then
\[ \mathcal{F}_0^\lambda V ={\rm span} \langle s_i \mid \lambda_{i,0} \geq\lambda \rangle \quad \text{ and } \quad \mathcal{F}_1^\lambda V ={\rm span} \langle s_i \mid \lambda_{i,1} \geq\lambda \rangle . \]
Using that 
$\dim _\kappa \mathcal {F}_0^{mx} V \cap \mathcal {F}_1^{my} V = \# \{ i \mid \lambda _{i,0} \geq mx \text { and }\lambda _{i,1} \geq my \}$, we compute
\[ \frac{ \partial^2}{\partial x \partial y } \dim_\kappa (\mathcal{F}_0^{mx} V \cap \mathcal{F}_1^{my} V) = \sum_{i=1}^{\ell} \delta_{ (m^{-1} \lambda_{i,0}, m^{-1} \lambda_{i, 1} )} . \]
Therefore, the restriction of 
$\mu _m$ to 
$(-\infty, N)\times (-\infty, N)$ equals
\begin{equation} \mu_m \big\vert_{ (-\infty,N)\times (-\infty,N)} = \frac{1}{m^n/n!} \sum_{ i \mid \lambda_{i,0} , \lambda_{i,1} < mN} \delta_{ (m^{-1} \lambda_{i,0}, m^{-1} \lambda_{i,1} )}. \end{equation}
Since 
$N>0$ was arbitrary, 
$\mu _m$ is a discrete measure on 
$\mathbb {R}^2$.
It remains to analyze the support of 
$\mu _m$. Note that 
$\mathcal {F}_0^{\lambda D} V \subset \mathcal {F}_1^{\lambda } V$ and 
$\mathcal {F}_1^{\lambda D} V \subset \mathcal {F}_0^\lambda V$ for all 
$\lambda \geq 1$. Therefore, the following statements hold.
(i) If
$\lambda _{i,1} >1$, then $\lambda _{i,0} \leq D \lambda _{i,1}$.(ii) If
$\lambda _{i,0} >1$, then $\lambda _{i,1} \leq D \lambda _{i,0}$.
Using (i), (ii), and (4.1), we see that 
$\mathrm {supp} (\mu _m) \subset \{ ({1}/{D}) x \leq y \leq Dx \} \cup \{ 0 \leq x \leq {1}/{m}\} \cup \{ 0 \leq y\leq {1}/{m}\}$.
Using (4.1), we show that 
$\mu _m$ encodes the colengths of 
$\mathfrak {a}_{\bullet,t}$.
Proposition 4.5 For each 
$t\in [0,1]$ and 
$a,b\in \mathbb {R}_{\geq 0}$,
\begin{align*} \mu_m\big( \{ (1-t)x+ty < a \} \big)& = \frac{\ell \big( R/ \mathfrak{a}_{ma,t}\big)}{m^n/n!}, \\ \mu_m\big( \{ x< a\} \cup \{ y< b\} \big) & = \frac{ \ell \big( R/ ( \mathfrak{a}_{ma,0} \cap \mathfrak{a}_{mb,1})\big)}{m^n/n!}. \end{align*}
Proof. By Lemma 4.4, it suffices to prove the result when 
$(R,\mathfrak {m})$ is complete. In this case we continue with the notation from the previous proof, but additionally fix 
$N>\max \{{a}/({1-t}), {b}/{t}\}$. Now, consider the filtration 
$\mathcal {F}^\bullet _t$ of 
$V$ defined by 
$\mathcal {F}_t^\lambda V:= {\rm im} ( \mathfrak {a}_{\lambda,t} \to V)$ for each 
$\lambda \in \mathbb {R}$. Observe that
\begin{equation} \mathcal{F}_t^{ma} V = \sum_{ ma = (1-t) \mu + t \nu } \mathcal{F}_0^\mu V \cap \mathcal{F}_1^\nu V = {\rm span} \langle s_i \mid (1-t)\lambda_{i,0} +t \lambda_{i,1} \geq ma \rangle, \end{equation}
where the sum runs through all 
$\mu, \nu \in \mathbb {R}$ satisfying 
$ma=(1-t)\mu +t \nu$. Now, we compute
\begin{align} \ell \big( R/\mathfrak{a}_{ma,t} \big) &= \dim_\kappa V / \mathcal{F}_t^{ma} V = \dim_\kappa V- \dim_\kappa \mathcal{F}_t^{ma} V \nonumber\\ &= \# \{ i \mid (1-t) \lambda_{i,0} + t\lambda_{i,1} < ma \} = \frac{m^n}{n!}\mu_m\big( \{ (1-t)x + ty < a\} \big) \end{align}
where the first equality uses that 
$\mathfrak {m}^{NCm} \subseteq \mathfrak {a}_{ma/(1-t),0} + \mathfrak {a}_{mb/t,1} \subset \mathfrak {a}_{ma,t}$, the third (4.2), and the fourth (4.1). Therefore, the first formula in the proposition holds.
To verify the second formula, we fix 
$N> \max \{a,b\}$ and similarly compute
\begin{align*} \dim_\kappa R/( \mathfrak{a}_{ma,0} \cap \mathfrak{a}_{mb,1} ) &= \dim_\kappa V / (\mathcal{F}_0^{ma} V \cap \mathcal{F}_1^{mb} V) = \dim_\kappa V- \dim_\kappa \mathcal{F}_0^{ma} V \cap \mathcal{F}_1^{mb} V\\ &= \# \{ j \mid \lambda_{i,0} < ma \text{ or } \lambda_{i,1} < mb \} = \frac{m^n}{n!}\mu_m\big( \{ x< a \} \cup \{y < b \} \big), \end{align*}
where the first equality uses that 
$\mathfrak {m}^{NCm} \subset \mathfrak {a}_{ma,0} \cap \mathfrak {a}_{mb,1}$.
4.3 Limit measure
We now construct a limit of the sequence of measures 
$(\mu _m)_{m \geq 1}$ that encodes the multiplicity along the geodesic.
Consider the function 
$H\colon \mathbb {R}^2 \to \mathbb {R}$ defined by
\[ H(x,y) = \mathrm{e}( \mathfrak{a}_{x\bullet ,0} \cap \mathfrak{a}_{y\bullet ,1} ) . \]
Above, we use the convention that 
$\mathrm {e}(R)=0$, which occurs when 
$x\leq 0$ and 
$y\leq 0$.
Proposition 4.6 The distributional derivative 
$\mu := - ({ \partial ^2 H}/{\partial x \partial y})$ is a measure on 
$\mathbb {R}^2$ and the sequence of measures 
$(\mu _m)_{m\geq 1}$ converges weakly to 
$\mu$ as 
$m\to \infty$.
Proof. We claim that 
$H_m$ converges to 
$H$ in 
$L^1_{ {\rm loc}}( \mathbb {R}^2)$. Assuming the claim, then 
$H_m$ converges to 
$H$ as distributions, and, hence, 
$\mu _m$ converges to 
$\mu := - ({ \partial ^2 H}/{\partial x \partial y})$ as distributions as well. Since each 
$\mu _m$ is a measure, [Reference HörmanderHör03, Theorem 2.1.9] implies 
$\mu$ is a measure and 
$\mu _m$ converges to 
$\mu$ weakly as measures.
It remains to prove the above claim. First, observe that 
$H_m$ converges to 
$H$ pointwise by the definition of the multiplicity as a limit. Next, fix an integer 
$N>0$. For 
$-N\leq x,y< N$, observe that
\[ 0 \leq H_m(x,y) \leq \frac{ \ell \big( R/ \mathfrak{m}^{N Cm} \big) }{ m^n/n! } \leq \sup_{m \geq 1} \frac{\ell \big( R/ \mathfrak{m}^{NCm}\big) }{ m^n/n! } <+\infty . \]
Therefore, the dominated convergence theorem implies that
\[ \lim_{m \to \infty} \int_{ (-N,N)\times (-N,N) } |H_m-H| =0. \]
Since 
$N>0$ was arbitrary, 
$H_m$ converges to 
$H$ in 
$L^1_{ {\rm loc}}( \mathbb {R}^2)$ as desired.
Proposition 4.7 For each 
$t\in [0,1]$ and 
$a,b\in \mathbb {R}_{\geq 0}$,
(1)
$a^n \mathrm {e}( \mathfrak {a}_{\bullet,t})=\mathrm {e}(\mathfrak {a}_{a\bullet,t}) =\mu ( \{ x(1-t)+yt < a \})$,(2)
$\mathrm {e}( \mathfrak {a}_{a\bullet,0 } \cap \mathfrak {a}_{b\bullet,1})= \mu \big ( \{ x< a\} \cup \{y< b\}\big )$.
Proof. Fix 
$\epsilon >0$. Since 
$(\mu _{m})_{m\geq 1}$ converges to 
$\mu$ weakly as measures by Proposition 4.6,
\begin{align*} \limsup \mu_m\big( \{ x(1-t) +y t< a-\epsilon \} \big) &\leq \mu\big( \{ x(1-t) +y t< a\}\big)\\ &\leq \liminf \mu_m \big( \{x(1-t)+yt < a\} \big) \end{align*}
Using Proposition 4.5(1) to compute the lim sup and lim inf, we deduce
\[ (a-\epsilon)^n\mathrm{e}(\mathfrak{a}_{\bullet,t}) = \mathrm{e}(\mathfrak{a}_{(a-\epsilon)\bullet,t}) \leq \mu\big( \{ x(1-t)+yt< a\}\big) \leq \mathrm{e}(\mathfrak{a}_{a\bullet,t})= a^n \mathrm{e}(\mathfrak{a}_{\bullet,t}). \]
Sending 
$\epsilon \to 0$ completes the proof of (1). The proof of the second is similar, but uses Proposition 4.5(2).
Lemma 4.8 For any real number 
$c>0$, the following statements are equivalent.
(1)
$\mathrm {supp}(\mu ) \subset \{ cx =y \}$.(2)
$\mathrm {e}(\mathfrak {a}_{\bullet,0}) = \mathrm {e}(\mathfrak {a}_{\bullet,0} \cap \mathfrak {a}_{c\bullet,1}) = \mathrm {e}(\mathfrak {a}_{c\bullet,1})$.
Proof. If (1) holds, then
\[ \mu\big(\{ x< 1\} \big) = \mu \big(\{ x <1\} \cup \{ y < c \} \big) = \mu\big( \{y< c \}\big). \]
Proposition 4.7 then implies (2) holds.
For the reverse implication, assume (2) holds. We claim
\[ H(x,y)= \begin{cases} \mathrm{e}(\mathfrak{a}_{x\bullet,0} ), & \text{if } cx\geq y,\\ \mathrm{e}(\mathfrak{a}_{ y\bullet,1} ), & \text{if } y\geq cx. \end{cases} \]
Indeed, if 
$cx\geq y$, then
\[ \mathrm{e}(\mathfrak{a}_{x\bullet ,0} ) \leq H(x,y) \leq \mathrm{e}(\mathfrak{a}_{x \bullet,0 } \cap \mathfrak{a}_{cx\bullet ,1}) = \mathrm{e}(\mathfrak{a}_{ x\bullet ,0}) . \]
Thus, the claim holds when 
$cx\geq y$ and similar reasoning treats the case when 
$cx< y$. Using the claim, we conclude (1) holds.
Proposition 4.9 The following statements hold.
(1) The support of
$\mu$ is contained in $\{({1}/{D}) x \leq y \leq D x\}$.(2)
$\mu$ is homogeneous of degree $n$ (i.e., $\mu (c\cdot A) = c^n \mu ( A)$ for any Borel set $A\subset \mathbb {R}^2$ and real number $c>0$).
Proof. By Propositions 4.3 and 4.6, 
$\mathrm {supp}(\mu ) \subset \{({1}/{D}) x \leq y \leq D x\} \cup \{ x=0\} \cup \{y =0 \}$. Note that
\[ \mu( \{x =0 \}) = \lim_{ a \to 0^+} \mu (\{ 0\leq x< a \} ) = \lim_{a \to 0^+} \mathrm{e}( \mathfrak{a}_{a \bullet,0}) = \lim_{a \to 0^+} a^n \mathrm{e}( \mathfrak{a}_{ \bullet,0}) = 0, \]
where the second equality is by Proposition 4.7. Similar reasoning shows 
$\mu (\{ y=0 \})$. Thus, (1) holds. Statement (2) follows from the fact that 
$H$ is homogeneous of degree 
$n$, meaning 
$H(cx,cy) = c^n H(x,y)$ for each 
$c>0$ and 
$(x,y) \in \mathbb {R}^2$.
4.4 Measure on the segment
To prove Theorem 1.1, it will be convenient to work with a measure induced by 
$\mu$ on the interval 
$[0,1]$.
Consider the embedding
\[ j\colon [0,1] \hookrightarrow \mathbb{R}^2 \quad \text{defined by}\quad j(z) := (z,1-z). \]
Note that 
$j([0,1])$ is the line segment between 
$(0,1)$ and 
$(1,0)$ in 
$\mathbb {R}^2$. For 
$A\subset [0,1]$, set
\[ {\rm Cone}(A) := \mathbb{R}_{\geq0} \cdot j(A) . \]
We define a measure 
$\widetilde {\mu }$ on 
$[0,1]$ by setting
\[ \widetilde{\mu}( A) := \mu \big( {\rm Cone}(A) \cap \{ x+ y < 1 \} \big) \]
for any Borel set 
$A \subset [0,1]$. Note that 
$\widetilde {\mu }$ is indeed a measure, since 
$\mu$ is a measure by Proposition 4.6 and 
$\mu ( \{{\bf 0}\} )=0$, which follows from Proposition 4.7. Additionally, 
$\mathrm {supp} (\widetilde {\mu })\subset [{1}/({D+1}), {D}/({D+1}) ]$, since 
$\mathrm {supp}(\mu )\subset \{ ({1}/{D}) x \leq y \leq Dx \}$.
Lemma 4.10 For any real number 
$c>0$, the following statements are equivalent.
(1)
$\mathrm {supp}(\tilde {\mu }) = {1}/({c+1})$.(2)
$\mathrm {e}(\mathfrak {a}_{\bullet, 0}) = \mathrm {e}(\mathfrak {a}_{\bullet,0} \cap \mathfrak {a}_{c\bullet,1}) = \mathrm {e}(\mathfrak {a}_{c\bullet, 1})$.
Proof. Using that 
$\mathbb {R}\cdot j ({1}/({c+1}))= \{cx=y \}$ and the definition of 
$\widetilde {\mu }$, we see that (1) holds if and only if 
${\rm supp}(\mu \vert _{\{0\leq x+y<1\}}) \subset \{cx=y \}$. Since 
$\mu$ is homogeneous by Proposition 4.9(2), 
${\rm supp} (\mu \vert _{\{0\leq x+y<1\}}) \subset \{xc=y\}$ if and only if 
${\rm supp}(\mu ) \subset \{cx=y\}$. By Lemma 4.8, the latter condition holds if and only if (2) holds.
Lemma 4.11 For 
$t\in [0,1]$,
\[ \mathrm{e}(\mathfrak{a}_{\bullet,t} ) = \int \big( (1-t)z +t(1-z) \big)^{-n} \, {d} \widetilde{\mu} . \]
Proof. For simplicity, we assume 
$t \leq \tfrac {1}{2}$, which ensures 
$g_t(z): =(1-t)z +t(1-z)$ is non-decreasing. (The case when 
$t>\tfrac {1}{2}$ can be proved similarly, but using left-hand approximations below.) For each 
$m\geq 1$, consider the simple function
\[ g_{t,m}(z) = \sum_{k=1}^{2^m} g_{t}(k/2^m) \cdot 1_{I_{k,m} } \]
where 
$I_{1,m} := [0,{1}/{2^m}]$ and 
$I_{k,m}:= ( ({k-1})/{2^m}, {k}/{2^m}]$ when 
$1< k\leq 2^m$. Since 
$g_{t,m} \geq g_{t,m+1}$ and 
$g_{t,m}\to g$ pointwise as 
$m\to \infty$, the monotone convergence theorem implies
\[ \int g_t(z)^{-n} \, {d} \widetilde{\mu} = \lim_{m \to \infty} \int g_{t,m}(z)^{-n} \, {d} \widetilde{\mu}. \]
(Above we are using that 
$g_t(z)^{-n}$ is defined and continuous on 
$(z,t) \in \mathrm {supp} (\widetilde {\mu }) \times [0,1]$).) Now, set 
$A_{k,m} := g_{t}(k/2^m)^{-1}\cdot ( {\rm Cone}(I_{k,m}) \cap \{x+y < 1\} ) \subset \mathbb {R}^2$ and 
$A_m = \cup _{k=1}^{2^m} A_{k,m}$. We compute
\[ \int g_{t,m}(z)^{-n} \, {d} \widetilde{\mu} = \sum_{k=1}^{2^m} g_{t}(k/2^m)^{-n} \widetilde{\mu}(I_{k,m}) = \sum_{k=1}^{2^m} \mu\big( A_{k,m} \big) = \mu(A_m) , \]
where the second equality uses Proposition 4.9(2). Next, notice that 
$A_{m} \subset A_{m+1}$ and
\begin{align*} \bigcup_{m \geq 1} A_m &= \bigcup_{m \geq 1} \bigcup_{z\in [0,1] }g_{t,m}(z)^{-1} \cdot ({\rm Cone}(\{z\}) \cap \{ x+y <1 \})\\ &= \bigcup_{z\in [0,1]} g_{t}(z)^{-1} \cdot ({\rm Cone}(\{z\}) \cap \{x+y <1 \}) = \mathbb{R}^2_{>0} \cap \{(1-t) x+ty <1 \}. \end{align*}
Combining the previous two computations, we conclude
\[ \int g_t(z)^{-n} \, {d} \widetilde{\mu}= \lim_{m\to \infty} \mu(A_m) = \mu \big( \mathbb{R}^2_{>0} \cap \{ (1-t)x+ ty <1 \} \big) = \mathrm{e}(\mathfrak{a}_{\bullet,t}) , \]
where the final equality is Proposition 4.7(1).
4.5 Proof of Theorem 1.1
Proof of Theorem 1.1 Set 
$g(z,t) := (1-t)z + t(1-z)$. By Lemma 4.11,
\[ E(t) = \int g(z,t)^{-n} \, {d}\widetilde{\mu} . \]
Since 
$g(z,t) = 0$ if and only if 
$z= {t}/({2t-1})$, there exists 
$\epsilon >0$ such that
\[ g(z,t) \neq 0\quad \text{for all } (z,t) \in \biggl[ \frac{1}{D+1}-\epsilon,\frac{D}{D+1} +\epsilon\biggr] \times [-\epsilon, 1+\epsilon] . \]
Since 
$g(z,t)$ is also smooth on 
$\mathbb {R}^2$, 
$g(z,t)^{-n}$ is smooth and has bounded derivatives on 
$({1}/({D+1})-\epsilon,{D}/({D+1})+\epsilon ) \times (-\epsilon, 1+\epsilon )$. Using that 
$\mathrm {{\rm supp}}\,{\tilde {\mu } } \subset [{1}/({D+1}),{D}/({D+1})]$, the Leibniz integral formula implies 
$\int g(z,t)^{-n}\, {d} \widetilde {\mu }$ is smooth on 
$(-\epsilon,1+\epsilon )$, which proves (1). The formula additionally implies
\[ E'(t)= n \int (2z-1)g(z,t)^{-n-1}\, {d} \tilde{\mu} \quad \text{and} \quad E''(t) = n(n+1) \int (2z-1)^2g(z,t)^{-n-2} \, {d} \tilde{\mu}. \]
To prove (2), it suffices to show 
$({d^2}/{dt^2}) (E(t)^{-1/n} )\leq 0$ for all 
$t\in (0,1)$. Note that
\[ \frac{d^2}{dt^2} \big(E(t)^{-1/n} \big) = \frac{(n+1) E'(t)^2- n E(t)E''(t) }{ n^2E(t)^{1/n+2} } . \]
To show the latter quantity is non-positive, we compute
\begin{align} nE(t) E''(t) &= n^2(n+1) \biggl( \int \big| g(z,t)^{-n} \big| \, {d} \tilde{\mu} \biggr) \biggl( \int \big| (2z-1)^2{g(z,t)^{-n-2}} \big| \, {d} \tilde{\mu} \biggr) \nonumber\\ &\geq n^2 (n+1) \biggl( \int \big| (2z-1) g(z,t)^{-n-1} \big| \, {d} \tilde{\mu}\biggr)^2 \geq (n+1) E'(t)^2 , \end{align}
where the first inequality is by the Cauchy–Schwarz inequality. Therefore, 
$({d^2}/{dt^2}) (E(t)^{-1/n}) \leq 0$ for all 
$t\in (0,1)$, which proves (2).
To prove (3), note that 
$E(t)$ is linear if and only if 
$({d^2}/{dt^2}) (E(t)^{-1/n} )=0$ for all 
$t\in (0,1)$. The latter holds if and only if the inequalities in (4.4) are all equalities. The inequalities are equalities if and only if 
$(2z-1)g(z,t)^{-(n+2)/2}$ and 
$g(z,t)^{-n/2}$ are linearly dependent in 
$L^{1}(\widetilde {\mu })$. Equivalently, 
$2z-1$ and 
$g(z,t)$ are linearly dependent in 
$L^{1} (\widetilde {\mu })$. The linear dependence holds exactly when 
$\widetilde {\mu }$ is supported at a single point. By Lemma 4.10 and Corollary 3.17, the latter condition is equivalent to the existence of 
$c\in \mathbb {R}$ such that 
$\widetilde {\mathfrak {a}}_{\bullet, 0}=\widetilde {\mathfrak {a}}_{c\bullet,1}$.
5. Applications
In this section, we prove Corollaries 1.2, 1.3, and 1.5 using results from previous sections.
5.1 Convexity of the volume of a valuation
Proof of Corollary 1.2 Let 
$(\mathfrak {a}_{\bullet,t})_{t\in [0,1]}$ denote the geodesic between 
$\mathfrak {a}_{\bullet,0}:= \mathfrak {a}_\bullet (v_{\boldsymbol {\alpha }})$ and 
$\mathfrak {a}_{\bullet,1}:= \mathfrak {a}_{\bullet }(v_{\boldsymbol {\beta }})$. Since 
$v_{(1-t) \boldsymbol {\alpha } + t \boldsymbol {\beta }}(f) \geq (1-t) v_{\boldsymbol {\alpha }} (f) + tv_{\boldsymbol {\beta }} (f)$ for any 
$f\in \mathcal {O}_{X,x}$, 
$\mathfrak {a}_{\bullet,t}\subset \mathfrak {a}_\bullet (v_{(1-t)\boldsymbol {\alpha } + t \boldsymbol {\beta }})$. We compute
\begin{align*} \mathrm{vol}(v_{(1-t)\boldsymbol {\alpha} + t \boldsymbol {\beta}})^{-1/n}&:= \mathrm{e}(\mathfrak{a}_\bullet((v_{(1-t)\boldsymbol {\alpha} + t \boldsymbol {\beta}}))^{-1/n} \geq \mathrm{e}(\mathfrak{a}_{\bullet,t})^{-1/n}\\ &\geq (1-t)\mathrm{e}(\mathfrak{a}_{\bullet,0})^{-1/n}+ t \mathrm{e}(\mathfrak{a}_{\bullet,1})^{-1/n} =(1-t)\,\mathrm{vol}(v_0)^{-1/n}+ t \,\mathrm{vol}(v_1)^{-1/n}, \end{align*}
where the second inequality is Theorem 1.1(2). Additionally, by Theorem 1.1(3) and Lemma 3.20, the second inequality is strict unless there exists 
$c\in \mathbb {R}_{>0}$ such that 
$\mathfrak {a}_{\bullet,0}= \mathfrak {a}_{c\bullet,1}$. Since 
$\mathfrak {a}_{\bullet,0}= \mathfrak {a}_{c\bullet,1}$ if and only if 
$c\boldsymbol {\alpha }_0 =\boldsymbol {\alpha }_1$, the result follows.
5.2 Uniqueness of normalized volume minimizer
Throughout this section, we assume 
$R$ is essentially of finite type over a field of characteristic 0.
To prove Corollary 1.3, we need the following proposition, which will be deduced from [Reference Xu and ZhuangXZ21, Theorem 3.11].
Proposition 5.1 Let 
$x\in (X,D)$ be a klt singularity. If 
$\mathfrak {a}_{\bullet,0}$ and 
$\mathfrak {a}_{\bullet,1}$ are 
$\mathfrak {m}$-filtrations, then
\[ \mathrm{lct}(X,D;\mathfrak{a}_{\bullet,t}) \leq (1-t) \cdot \mathrm{lct}(X,D;\mathfrak{a}_{\bullet,0})+ t \cdot \mathrm{lct}(X,D;\mathfrak{a}_{\bullet,1}), \]
for all 
$t\in (0,1)$.
Proof. We claim 
$v(\mathfrak {c}_\bullet )= v(\mathfrak {a}_{ \bullet,t} )$ for all 
$v\in \mathrm {Val}_{R,\mathfrak {m}}$, where 
$\mathfrak {c}_\bullet$ is the 
$\mathfrak {m}$-filtration defined by 
$\mathfrak {c}_\lambda := \sum _{i=0}^{\lceil \lambda \rceil } (\mathfrak {a}_{(\lceil \lambda \rceil -i)/(1-t),0 } \cap \mathfrak {a}_{i/t,0})$. Assuming the claim holds,
\begin{align*} \mathrm{lct}(X,D;\mathfrak{a}_{\bullet, t} ) &=\mathrm{lct}(X,D;\mathfrak{c}_\bullet)\\ &\leq \mathrm{lct}(X,D;\mathfrak{a}_{\bullet/(1-t),0}) + \mathrm{lct}(X,D;\mathfrak{a}_{\bullet/t,1}) = (1-t) \mathrm{lct}(X,D;\mathfrak{a}_{\bullet,0}) + t \mathrm{lct}(X,D;\mathfrak{a}_{\bullet,1}), \end{align*}
where the inequality is [Reference Xu and ZhuangXZ21, Theorem 3.11]. It remains to verify the claim. Since 
$\mathfrak {c}_{m} \subset \mathfrak {a}_{t,m}$ for each integer 
$m>0$, 
$v(\mathfrak {c}_{\bullet }) \geq v(\mathfrak {a}_{t,\bullet })$ for all 
$v\in \mathrm {Val}_{R,\mathfrak {m}}$. Next fix 
$\mu, \nu \in \mathbb {R}$ such that 
${\mu (1-t)+\nu t = m}$. Set 
$r:= \mu (1-t) = m - \nu t$. For each 
$\ell \in \mathbb {Z}_{>0}$,
\begin{align*} (\mathfrak{a}_{\mu,0} \cap
\mathfrak{a}_{\nu,1})^\ell \subset ( \mathfrak{a}_{\ell
\mu,0} \cap \mathfrak{a}_{\ell \nu,1}) &=
(\mathfrak{a}_{\ell r/(1-t),1} \cap \mathfrak{a}_{\ell
(m-r)/t,1}) \subset (\mathfrak{a}_{\lfloor \ell r \rfloor
/(1-t),0}\\ &\quad \cap \mathfrak{a}_{\big(\ell m -1 - \lfloor \ell
r \rfloor \big)/t,1} ) \subset \mathfrak{c}_{\ell m -1
}. \end{align*}
Using the previous inclusions, we see that
\[ v(\mathfrak{a}_{\mu,0} \cap \mathfrak{a}_{\nu,1}) = \lim_{\ell \to \infty} \frac{v( (\mathfrak{a}_{\mu,0} \cap \mathfrak{a}_{\nu,1})^\ell)}{ \ell} \geq \lim_{\ell\to \infty} \frac{v(\mathfrak{c}_{\ell m-1}) }{ \ell} = v(\mathfrak{c}_\bullet) . \]
We can now compute
\[ v(\mathfrak{a}_{\bullet,t}) = \lim_{m\to \infty} \frac{v(\mathfrak{a}_{m,t})}{m} = \lim_{m\to \infty} \min_{ \mu(1-t)+\nu t= m} \biggl\{ \frac{v(\mathfrak{a}_{\mu,0} \cap \mathfrak{a}_{\nu,1})}{m} \biggr\} \geq v(\mathfrak{c}_\bullet) . \]
Therefore, 
$v(\mathfrak {a}_{\bullet,t})= v(\mathfrak {c}_\bullet )$ as desired.
Proof of Corollary 1.3 Let 
$v_0,v_1 \in \mathrm {Val}_{R,\mathfrak {m}}$ be minimizers of 
$\widehat {\mathrm {vol}}_{(X,D),x}$. By rescaling, we may assume 
$A_{X,\Delta }(v_i) = 1$ and, hence, 
$\mathrm {vol}(v_i)= \widehat {\mathrm {vol}}(x,X,D)$ for 
$i=0,1$. We seek to show 
$v_0=v_1$.
Consider the geodesic 
$(\mathfrak {a}_{t,\bullet })_{t\in [0,1]}$ between 
$\mathfrak {a}_{\bullet,0} := \mathfrak {a}_{\bullet }(v_0)$ and 
$\mathfrak {a}_{\bullet,1}:= \mathfrak {a}_{\bullet }(v_1)$. For 
$t\in (0,1)$,
\begin{align} \mathrm{lct}(X,D; \mathfrak{a}_{\bullet,t}) &\leq (1-t) \mathrm{lct}(X,D;\mathfrak{a}_{\bullet,0}) + t \mathrm{lct}(X,D;\mathfrak{a}_{\bullet,1})\nonumber\\ &\leq (1-t) A_{X,D}(v_0) + t A_{X,D}(v_1)=1 , \end{align}
where the first inequality is Proposition 5.1 and the second uses that 
$v_i(\mathfrak {a}_{\bullet,i})=1$ by [Reference BlumBlu18, Lemma 3.5]. Observe that
\begin{align*} \widehat{\mathrm{vol}}(x,X,D)^{1/n} &\leq \mathrm{lct}(\mathfrak{a}_{\bullet,t}) \mathrm{e}(\mathfrak{a}_{\bullet,t})^{1/n} \leq \mathrm{e}(\mathfrak{a}_{\bullet,t})^{1/n} \\ &\leq \big( (1-t) \mathrm{e}(\mathfrak{a}_{\bullet,0})^{-1/n}+ t\mathrm{e}(\mathfrak{a}_{\bullet,1})^{-1/n}\big)^{-1} =\widehat{\mathrm{vol}}(X,D,x)^{1/n} \end{align*}
where there first inequality holds by [Reference LiuLiu18, Theorem 27], the second by (5.1), and the third by Theorem 1.1(2). Therefore, every inequality above, in particular the third one, is an equality. By Theorem 1.1(3) and Lemma 3.20, there exists 
$c>0$ such that 
$\mathfrak {a}_{\bullet,0}= \mathfrak {a}_{c\bullet,1}$, which translates to 
$\mathfrak {a}_{\bullet }(cv_0)= \mathfrak {a}_{\bullet }(v_1)$. Therefore, 
$cv_0=v_1$. Since 
$c=A_{X,D}(cv_0)=A_{X,D}(v_1) = 1$, we see that 
$v_0=v_1$.
Remark 5.2 [Reference LiuLiu18, Theorem 27] was originally proved for 
$\mathbb {Q}$-coefficients, but the proof works for 
$\mathbb {R}$-coefficients with little change.
5.3 Minkowksi inequality
Lastly, we prove Corollary 1.5. In the proof we use the following result.
Proposition 5.3 [Reference CutkoskyCut21, Theorem 10.3]
Let 
$\mathfrak {a}_\bullet$ and 
$\mathfrak {b}_\bullet$ be 
$\mathfrak {m}$-filtrations with positive multiplicity. If 
$\mathrm {e}(\mathfrak {a}_\bullet \mathfrak {b}_\bullet )^{1/n}= \mathrm {e}(\mathfrak {a}_\bullet )^{1/n}+\mathrm {e}(\mathfrak {b}_\bullet )^{1/n}$, then there exists 
$c>0$ such that 
$v(\mathfrak {a}_\bullet )=c v(\mathfrak {b}_\bullet )$ for all 
$v\in \mathrm {DivVal}_{R,\mathfrak {m}}$.
Proof of Corollary 1.5 By [Reference CutkoskyCut15, Theorem 3.2],
\begin{equation} \mathrm{e}(\mathfrak{a}_\bullet\mathfrak{b}_\bullet)^{1/n}\leq \mathrm{e}(\mathfrak{a}_\bullet)^{1/n}+\mathrm{e}(\mathfrak{b}_\bullet)^{1/n} . \end{equation}Thus, it remains to analyze when the inequality is an equality.
If (5.2) is an equality, then Propositions 5.3 and 3.9 imply there exists 
$c>0$ such that 
$\widetilde {\mathfrak {a}}_\bullet = \widetilde {\mathfrak {b}}_{c \bullet }$. Conversely, assume there exists 
$c>0$ such that 
$\widetilde {\mathfrak {a}}_\bullet = \widetilde {\mathfrak {b}}_{c \bullet }$. Note that
\[ v(\mathfrak{a}_\bullet \mathfrak{b}_\bullet) = v(\mathfrak{a}_\bullet) + v(\mathfrak{b}_\bullet) = (c+1) v(\mathfrak{b}_\bullet) \]
where the first equality uses that 
${ v(\mathfrak {a}_m \mathfrak {b}_m) }/{m} = {v(\mathfrak {a}_m)}/{m} + {v(\mathfrak {b}_m)}/{m}$ for all 
$m\in \mathbb {N}$ and the second is by Proposition 3.9. Therefore, Proposition 3.9 implies 
$\widetilde {\mathfrak {a}_\bullet \mathfrak {b}_\bullet }= \widetilde {\mathfrak {b}}_{(c+1)\bullet }$. We now compute
\[ \mathrm{e}(\mathfrak{a}_\bullet \mathfrak{b}_\bullet)^{1/n} = \mathrm{e}(\mathfrak{b}_{(c+1)\bullet})^{1/n} = \mathrm{e}(\mathfrak{b}_{ c \bullet})^{1/n} + \mathrm{e}(\mathfrak{b}_\bullet)^{1/n} = \mathrm{e}(\mathfrak{a}_\bullet)^{1/n} + \mathrm{e}(\mathfrak{b}_\bullet)^{1/n} , \]
where the first and third equality hold by Corollary 3.17.
6. Relation to global results
In this section, we explain the relationship between the local constructions in this paper and certain global constructions in the K-stability literature.
Throughout, let 
$X$ be an 
$n$-dimensional normal projective variety over a field 
$k$ and 
$L$ be an ample line bundle on 
$X$. The section ring of 
$(X,L)$ is
\[ R(X,L):= \bigoplus_{m\in \mathbb{N}} R_m := \bigoplus_{m \in \mathbb{N}} H^0(X,mL) . \]
6.1 Filtrations and multiplicity
The analogue of an 
$\mathbb {R}$-filtration of a local ring in the global setting is the following definition [Reference Boucksom, Hisamoto and JonssonBHJ17, § 1.1], which plays an important role in the K-stability literature.
Definition 6.1 A filtration 
$\mathcal {F}$ of 
$R(X,L)$ is the data of vector subspaces 
$\mathcal {F}^\lambda R_m \subset R_m$ for each 
$\lambda \in \mathbb {R}$ and 
$m\in \mathbb {N}$ such that
(1)
$\mathcal {F}^\lambda R_m \subset \mathcal {F}^\mu R_m$ when $\lambda >\mu$,(2)
$\mathcal {F}^\lambda R_m = \mathcal {F}^{\lambda -\epsilon } R_m$ when $0 <\epsilon \ll 1$,(3)
$\mathcal {F}^\lambda R_m \cdot \mathcal {F}^\mu R_n \subset \mathcal {F}^{\lambda +\mu }$,(4)
$\mathcal {F}^{\lambda } R_m = R_m$ when $\lambda \ll 0$, and(5)
$\mathcal {F}^{\lambda } R_m =0$ when $\lambda \gg 0$.
A filtration 
$\mathcal {F}$ is linearly bounded if there exists 
$C>0$ such that 
$\mathcal {F}^{Cm}R_m=0$ for all 
$m>0$.
The data of a filtration 
$\mathcal {F}$ can be encoded as a norm 
$\chi : R(X,L)\to \mathbb {R}\cup \{ +\infty \}$ by setting
\[ \chi (s) := \max \{ \lambda \in \mathbb{R} \mid s \in \mathcal{F}^\lambda R_m \} \]
when 
$s \in R_m$ and 
$\chi (\sum _m s_m ):= \min \{ \chi (s_m ) \}$ when 
$\sum _m s_m \in \oplus _{m} R_m$; see [Reference Boucksom and JonssonBJ21, § 1.1] for details. Following [Reference Boucksom and JonssonBJ21, § 1.1], we write 
$\mathcal {N}_{\mathbb {R}}$ for the set of such norms 
$\chi : R(X,L) \to \mathbb {R} \cup \{ +\infty \}$ that arise from linearly bounded 
$\mathbb {R}$-filtrations of 
$R(X,L)$.
Definition 6.2 The energy of a linearly bounded filtration 
$\mathcal {F}$ of 
$R(X,L)$ is
\[ E (\mathcal{F}) := \lim_{m \to \infty} \frac{ \sum_{\lambda \in \mathbb{R}}\dim_k \mathrm{gr}_{\mathcal{F}}^\lambda R_m }{m^n / n!}, \]
where 
$\mathrm {gr}_{\mathcal {F}}^\lambda R_m = \mathcal {F}^\lambda R_m / \mathcal {F}^{\lambda +\epsilon }R_m$ and 
$0<\epsilon \ll 1$. The limit in the definition exists as a consequence [Reference Boucksom, Hisamoto and JonssonBHJ17, Theorem 5.3].
The energy measures the size of a filtration and is an analogue of the multiplicity of an 
$\mathbb {R}$-filtration in the global setting. The invariant appears under different names in the K-stability literature such as the energy in [Reference Boucksom, Hisamoto and JonssonBHJ17] and the 
$S$-invariant in [Reference XuXu21]. For a norm 
$\chi \in \mathcal {N}_{\mathbb {R}}$ of 
$R(X,L)$, 
$\mathrm {vol}(\chi )$ denotes the energy of the associated filtration in [Reference Boucksom and JonssonBJ21].
6.2 Saturation and Rees's theorem
Rees's theorem for 
$\mathbb {R}$-filtrations (Theorem 1.4) says that two 
$\mathbb {R}$-filtrations 
$\mathfrak {a}_\bullet \subset \mathfrak {b}_\bullet$ have equal volume if and only if their saturations are equal. As we will explain, an analogue of this result was previously proven by Boucksom and Jonsson in the global setting.
For a norm 
$\chi \in \mathcal {N}_{\mathbb {R}}$, Boucksom and Jonsson introduce the notion of its maximal norm 
$\chi ^{\max }$ [Reference Boucksom and JonssonBJ21, Definition 6.16]. Similar to the saturation in Definition 3.1, it is defined using divisorial valuations. When 
${\rm char}(k)=0$, they prove that two norms 
$\chi, \chi ' \in \mathcal {N}_{\mathbb {R}}$ with 
$\chi \leq \chi '$ satisfy 
$\mathrm {vol}(\chi ) = \mathrm {vol}(\chi ')$ if and only if 
$\chi ^{\max }= \chi '^\max$ [Reference Boucksom and JonssonBJ21, Lemma 3.11 and Theorem 6.22]. The proof relies on results from non-Archimedean pluripotential developed in [Reference Boucksom and JonssonBJ22].
6.3 Geodesics
The notion of a geodesic between two 
$\mathbb {R}$-filtrations (Definition 4.1) is inspired by the following definition in the global setting.
Definition 6.3 [Reference Blum, Liu, Xu and ZhuangBLXZ23, Reference RebouletReb22]
For two linearly bounded filtrations 
$\mathcal {F}_0$ and 
$\mathcal {F}_1$ of 
$R(X,L)$ and 
$t\in (0,1)$, we define a filtration 
$\mathcal {F}_t$ of 
$R(X,L)$ by setting
\[ \mathcal{F}_t^\lambda R_m= \sum_{\mu+\nu =\lambda} \mathcal{F}_0^\mu R_m \cap \mathcal{F}_1^\nu R_m . \]
We call 
$(\mathcal {F}_t)_{t\in [0,1]}$ the geodesic between 
$\mathcal {F}_0$ and 
$\mathcal {F}_1$.
This definition was introduced by the first two authors, Xu, and Zhuang in [Reference Blum, Liu, Xu and ZhuangBLXZ23, § 3.1.2] to prove uniqueness results for certain optimal destabilizations of Fano varieties that arise from limits of Kähler–Ricci flow [Reference Blum, Liu, Xu and ZhuangBLXZ23, § 3]. Independently, Reboulet introduced an equivalent definition phrased in the language of norms on the section ring, rather than filtrations, and used it to define geodesics between metrics on a line bundle in non-Archimedean pluripotential theory [Reference RebouletReb22].
A different and possibly more intuitive way to understand the above definition is in terms of a well-chosen basis. By [Reference Abban and ZhuangAZ22, Lemma 3.1], there exists a basis 
$(s_1,\ldots, s_{N_m}\!)$ of 
$R_m$ that is compatible with both 
$\mathcal {F}_0$ and 
$\mathcal {F}_1$. Here, ‘compatible’ means that 
$\mathcal {F}_j^\lambda R_m$ is the span of some subset of 
$(s_1,\ldots,s_{N_m}\!)$ for each 
$j \in \{0,1\}$ and 
$\lambda \in \mathbb {R}$. If we set 
$\lambda _{i,j} := \max \{ \lambda \in \mathbb {R}\mid s_i \in \mathcal {F}_j^\lambda R_m \}$, then a computation shows
\[ \mathcal{F}_t^\lambda R_m := {\rm span}( s_i \mid \lambda_{i,0}(1-t)+ \lambda_{i,1} t \geq \lambda) . \]
(A similar computation is made in the proof of Proposition 4.3.) Imprecisely, this expression shows that 
$(\mathcal {F}_{t})_{t\in [0,1]}$ interpolates between 
$\mathcal {F}_0$ and 
$\mathcal {F}_1$.
As shown in [Reference Blum, Liu, Xu and ZhuangBLXZ23, § 3.2] and [Reference RebouletReb23, § 4.7], various functionals on the space of filtrations are convex along geodesics. These convexity results are proven in [Reference Blum, Liu, Xu and ZhuangBLXZ23, § 3.2] using a measure on 
$\mathbb {R}^2$ (similarly to the proof of Theorem 1.1) and imply the uniqueness of minimizers of the 
$h$-functional on the space of valuations on a Fano variety [Reference Blum, Liu, Xu and ZhuangBLXZ23, § 3.3]. Thus, Theorem 1.1 and Corollary 1.3 can be viewed as local counterparts to these global results.
Acknowledgements
We would like to thank Mattias Jonsson, Linquan Ma, Sam Payne, Longke Tang, Chenyang Xu, and Ziquan Zhuang for helpful discussions. LQ would like to thank his advisor Chenyang Xu for constant support, encouragement and numerous inspiring conversations. We are also grateful to the referees whose comments and corrections improved the quality of this paper. HB is partially supported by NSF grant DMS-2200690. YL is partially supported by NSF Grant DMS-2148266.
Conflicts of Interest
None.
Appendix A. Okounkov bodies and geodesics
In this appendix, we give an alternative proof of a weaker version of Theorem 1.1 (see Theorem A.7) using the theory of Okounkov bodies.
Throughout, 
$X$ is an 
$n$-dimensional normal variety over an algebraically closed field 
$\mathbb {k}$ and 
$x\in X$ is a closed point. We set 
$R: = \mathcal {O}_{X,x}$ and write 
$\mathfrak {m}\subset R$ for the maximal ideal. Note that the assumptions on 
$R$ are more restrictive than in the earlier sections of this paper.
A.1 Good valuations
Definition A.1 [Reference Kaveh and KhovanskiiKK14, Definition 8.3]
We equip a total order 
$\preceq$ on 
$\mathbb {Z}^n$ respecting addition. Let 
$v: R\setminus \{0\}\to \mathbb {Z}^n$ be a valuation. We say that 
$v$ is a good valuation if it satisfies the following conditions.
(1)
$v$ has one-dimensional leaves, that is, for any $f,g\in R$ with $v(f)=v(g)$, there exists $\lambda \in \mathbb {k}$ such that $v(f+\lambda g)>v(f)$.(2) The value semigroup
$\mathcal {S}=v(R\backslash \{0\})\cup \{0\}$ generates $\mathbb {Z}^n$ as a group, and its associated cone $C(\mathcal {S})$, which is the closure of the convex hull of $\mathcal {S}$, is a strongly convex cone. The latter is equivalent to the existence of a linear function $\xi :\mathbb {R}^n\to \mathbb {R}$ such that $C(\mathcal {S})\backslash \{0\}\subset \xi _{> 0}$.(3) There exist
$r_0>0$ and a linear function $\xi$ as above such that
for any\[ \mathrm{ord}_\mathfrak{m}(f)\ge r_0\xi(v(f)) \]
$f\in R\backslash \{0\}$.
For every quasi-monomial valuation 
$w\in \mathrm {Val}_{X,x}$, we can construct a good valuation 
$v$ on 
$X$ associated to 
$w$ as follows. The construction is similar to [Reference Kaveh and KhovanskiiKK14, Example 8.5], where the characteristic of 
$\mathbb {k}$ is 
$0$. We provide a proof for the general case here for the reader's convenience.
Lemma A.2 If 
$w\in \mathrm {Val}_{X,x}$ is a quasi-monomial valuation, then there exist a vector 
$\boldsymbol{\alpha} \in \mathbb {R}^n$ and a good 
$\mathbb {Z}^n$-valued valuation 
$v$ on 
$R$ such that
\[ w(f)=\langle \boldsymbol{\alpha}, v(f) \rangle \]
for any 
$f\in R\backslash \{0\}$. In addition, we can take 
$\xi =\langle \boldsymbol{\alpha},\cdot \rangle$ for condition (3) of a good valuation.
Proof. Since 
$w$ is quasi-monomial, there is a birational morphism of varieties 
$\mu :Y\to X$, a point 
$\eta \in Y$, and a divisor 
$D$ on 
$Y$ that is simple normal crossing (snc) at 
$\eta$ such that 
$w\in \mathrm {QM}_{\eta }(Y,D)$. We may further assume 
$\eta \in Y$ is a closed point and 
$D$ is locally defined at 
$\eta$ by a regular system of a parameters 
$y_1,\ldots, y_n \in \mathcal {O}_{Y,\eta }$. (Indeed, since the locus where 
$(Y,D)$ is snc is open, we may replace 
$\eta$ with closed point in 
$\overline {\eta }$ and possibly add components to 
$D$ so that the latter holds.) By replacing 
$Y$ with a higher birational model as in the proof of [Reference Jonsson and MustaţăJM12, Lemma 3.6], we may assume 
$w=v_{\boldsymbol {\alpha }}$, where 
${\boldsymbol {\alpha }}:= (\alpha _1,\ldots, \alpha _r, 0, \ldots, 0)$ and 
$\alpha _1,\ldots, \alpha _r$ are linearly independent over 
$\mathbb {Q}$.
Choose 
$\alpha '_{r+1},\ldots,\alpha '_n>0$ linearly independent over 
$\mathbb {Q}$ and define 
$\boldsymbol{\alpha} ':= (0,\ldots,0,\alpha '_{r+1},\ldots,\alpha '_n)$. Then we can define a quasi-monomial valuation 
$u$ on 
$k[\![y_1,\ldots,y_n]\!]\cong \widehat{\mathcal {O}}_{Y,z}$ given by the weight vector 
$\boldsymbol{\alpha} '$. Define 
$\boldsymbol {u}:\widehat{\mathcal {O}}_{Y,z}\backslash \{0\}\to (\mathbb {R}^2,\le _{\rm lex})$ by
\[ \boldsymbol {u}(f):= (w(f),u(f)). \]
Then it is easy to check that 
$\boldsymbol {u}$ is an 
$\mathbb {R}^2$-valued valuation. Moreover, it induces a well-order 
$\preceq$ on 
$\mathbb {N}^n$ by
\[ \boldsymbol{\beta}_1\preceq \boldsymbol{\beta}_2 \text{ if and only if } (\langle \boldsymbol{\alpha},\boldsymbol{\beta}_1 \rangle, \langle \boldsymbol{\alpha}',\boldsymbol{\beta}_1 \rangle)\le_{\rm lex} (\langle \boldsymbol{\alpha},\boldsymbol{\beta}_2 \rangle, \langle \boldsymbol{\alpha}',\boldsymbol{\beta}_2 \rangle). \]
Now define 
$v:\widehat{\mathcal {O}}_{Y,z}\backslash \{0\}\to (\mathbb {N}^n,\preceq )$ by 
$v(f):= \min \{\boldsymbol{\beta} \mid c_{\boldsymbol{\beta} }\ne 0\}$. Then 
$v$ is a 
$\mathbb {Z}^n$-valued valuation on 
$X$ with center 
$x$.
Following the calculation in [Reference CutkoskyCut13, § 4], we see that the value semigroup 
$\mathcal {S}$ contains 
$\boldsymbol{\beta} '$ and 
$\boldsymbol{\beta} '+\boldsymbol {e}_i$ for some 
$\boldsymbol{\beta} '$ and 
$1\le i\le n$, where 
$\boldsymbol {e}_i\in \mathbb {Z}^n$ is the 
$i$th standard basis vector. Hence, 
$\mathcal {S}$ generates 
$\mathbb {Z}^n$ as a group.
By definition we have 
$w(f)=\langle \boldsymbol{\alpha},v(f) \rangle$. Let 
$\xi := \langle \boldsymbol{\alpha},\cdot \rangle :\mathbb {R}^n\to \mathbb {R}$. Then we know that 
$\xi (\boldsymbol{\beta} )\ge \min \{\alpha _i\mid 1\le i\le r\}>0$ for any 
$0\ne \boldsymbol{\beta} =v(f)\in \mathcal {S}$. So 
$C(\mathcal {S})\backslash \{0\}\subset \xi _{>0}$ and condition (2) is satisfied.
By Izumi's inequality Lemma 2.4, there exists 
$r>0$ such that 
$w(f)\le r\cdot \mathrm {ord}_\mathfrak {m}(f)$ for any 
$f\in R\backslash \{0\}$. So if we set 
$r_0:= r^{-1}$, then
\[ \mathrm{ord}_\mathfrak{m}(f)\ge r_0w(f)=r_0\xi(v(f)). \]
Thus, condition (3) is satisfied. Condition (1) is straightforward to verify. So we conclude 
$v$ is a good 
$\mathbb {Z}^n$-valued valuation.
A.2 A volume formula for convex bodies
The following lemma will be useful in proving the convexity of multiplicities. It is a slight generalization of [Reference IzmestievIzm14, Lemma 4.4] which follows from the same argument, so we omit the proof.
Lemma A.3 [Reference IzmestievIzm14]
Let 
$C\subset \mathbb {R}^n$ be a strongly convex cone of dimension 
$n$, and 
$h:\mathrm {Int}(C)\to \mathbb {R}_{\ge 0}$ a continuous function homogeneous of degree 
$1$. Then
\[ n!\,\mathrm{vol}(h_{\le 1})=\int_C e^{-h}\,d\mu, \]
where 
$h_{\le 1}:= \{x\in \mathrm {Int}(C)\mid h(x)\le 1\}$ and 
$\mu$ is the Lebesgue measure on 
$\mathbb {R}^n$.
A.3 The cutting function induced by a filtration
To proceed, we fix a quasi-monomial valuation 
$w\in \mathrm {Val}_{X,x}$ and a good 
$\mathbb {Z}^n$-valued valuation 
$v$ associated to 
$w$ given by Lemma A.2. Let 
$\mathcal {S}=v(R\backslash \{0\})\cup \{0\}\subset \mathbb {Z}^n$ be the value semigroup of 
$v$. Let 
$\mathfrak {a}_{\bullet }$ be a linearly bounded 
$\mathfrak {m}$-filtration. The goal of this section is to construct a cutting function on the strongly convex cone 
$C(\mathcal {S})$ satisfying the conditions in Lemma A.3.
The following lemma is an easy generalization of Izumi's inequality to filtrations. For a family version, see Lemma A.10.
Lemma A.4 With notation as above, for any 
$m\in \mathbb {R}_{\ge 0}$, there exists 
$\boldsymbol{\beta} _0\in \mathcal {S}$ such that if 
$f\in R\backslash \{0\}$ satisfies 
$v(f)\succeq \boldsymbol{\beta} _0$, then 
$f\in \mathfrak {a}_{>m}$.
Proof. By Lemma 2.4, there exists 
$r>0$ such that 
$\xi (v(f))=w(f)\le r\cdot \mathrm {ord}_\mathfrak {m}(f)$. By assumption, there exists 
$d\in \mathbb {Z}_{>0}$ such that 
$\mathfrak {m}^{d}\subset \mathfrak {a}_1$. So if we choose 
$\boldsymbol{\beta} _0\in \mathcal {S}$ such that 
$\xi (\boldsymbol{\beta} _0)\ge rd(\lceil m\rceil +1)$, then for 
$v(f)\succeq \boldsymbol{\beta} _0$, we have 
$\mathrm {ord}_\mathfrak {m}(f)\ge d(\lceil m\rceil +1)$. Hence,
\[ f\in\mathfrak{m}^{d(\lceil m\rceil +1)}\subset \mathfrak{a}_1^{\lceil m \rceil+1}\subset \mathfrak{a}_{>m}. \]
We define a function 
$m_{\mathfrak {a}_{\bullet }}:\mathcal {S}\to \mathbb {R}_{\geq 0}$ associated to 
$\mathfrak {a}_{\bullet }$ by
\begin{equation} m_{\mathfrak{a}_{\bullet}}(\boldsymbol{\beta}):= \sup\{m\in \mathbb{R}_{\geq 0}\mid v^{-1}(\boldsymbol{\beta})\cap \mathfrak{a}_m\ne \emptyset\} \end{equation}
for any 
$\boldsymbol{\beta} \in \mathcal {S}$. Since 
$\mathfrak {a}_\bullet$ is left continuous, the above supremum is indeed a maximum. Now we choose an element 
$f_{\mathfrak {a}_{\bullet },\boldsymbol{\beta} }\in v^{-1}(\boldsymbol{\beta} )\cap \mathfrak {a}_{m_{\mathfrak {a}_{\bullet }}(\boldsymbol{\beta} )}$. We will write 
$m(\boldsymbol{\beta} )$ and 
$f_{\boldsymbol{\beta} }$ respectively, if there is no chance of ambiguity.
We first prove some properties of 
$m(\cdot )$ and construct a function 
$h$ on 
$\mathcal {S}$ using it, which will play a key role in estimating the multiplicities.
Proposition A.5 Let 
$m=m_{\mathfrak {a}_\bullet }:\mathcal {S}\to \mathbb {R}_{\geq 0}$ be the function associated to a linearly bounded 
$\mathfrak {m}$-filtration 
$\mathfrak {a}_{\bullet }$ as above. Then the following statements hold.
(1)
$m$ is superadditive, that is,
for any\[ m(\boldsymbol{\beta}_1+\boldsymbol{\beta}_2)\ge m(\boldsymbol{\beta}_1)+m(\boldsymbol{\beta}_2), \]
$\boldsymbol{\beta} _1,\boldsymbol{\beta} _2\in \mathcal {S}$.(2) There exists
$M_2>0$ such that for any $\boldsymbol{\beta} \in \mathcal {S}$, we have $m(\boldsymbol{\beta} )\le M_2\xi (\boldsymbol{\beta} )$.(3) If
$\mathfrak {a}_\bullet =\mathfrak {a}_\bullet (u)$ for some valuation $u\in \mathrm {Val}_{X,x}$, then we have
for all\[ u(f)\le\lim_{k\to\infty} \frac{m(kv(f))}{k} \]
$f\in R\setminus \{0 \}$, and the equality holds if $u=w$.(4) There is a continuous, concave function
$h_{\mathfrak {a}_{\bullet }}:\mathrm {Int}(C(\mathcal {S}))\to \mathbb {R}_{\ge 0}$ homogeneous of degree $1$, such that for any $\boldsymbol{\beta} \in \mathcal {S}$, $h_{\mathfrak {a}_{\bullet }}(\boldsymbol{\beta} )=\lim _{k\to \infty }({m_{\mathfrak {a}_{\bullet }}(k\boldsymbol{\beta} )}/{k})$. As before, we will drop $\mathfrak {a}_{\bullet }$ from the notation of $h(\boldsymbol{\beta} )$ if there is no ambiguity.(5) For
$\lambda >0$, we have
(A.2)\begin{equation} h_{\mathfrak{a}_{\lambda\bullet}}=h_{\mathfrak{a}_\bullet}/\lambda. \end{equation}
Proof. (1) For any 
$\boldsymbol{\beta} _1,\boldsymbol{\beta} _2\in \mathcal {S}$, we have
\[ f_{\boldsymbol{\beta}_1}f_{\boldsymbol{\beta}_2}\in v^{-1}(\boldsymbol{\beta}_1+\boldsymbol{\beta}_2)\cap \mathfrak{a}_{m(\boldsymbol{\beta}_1)}\mathfrak{a}_{m(\boldsymbol{\beta}_2)}\subset v^{-1}(\boldsymbol{\beta}_1+\boldsymbol{\beta}_2)\cap \mathfrak{a}_{m(\boldsymbol{\beta}_1)+m(\boldsymbol{\beta}_2)}. \]
Hence, 
$m(\boldsymbol{\beta} _1+\boldsymbol{\beta} _2)\ge m(\boldsymbol{\beta} _1)+m(\boldsymbol{\beta} _2)$.
(2) By Lemma 2.4 there exists 
$r>0$ such that 
$\mathrm {ord}_\mathfrak {m}(f)< rw(f)=r\xi (v(f))$. Hence, if we choose 
$f\in v^{-1}(\boldsymbol{\beta} )$, then we know that 
$\mathrm {ord}_\mathfrak {m}(f)< r\xi (\boldsymbol{\beta} )$, that is, 
$f\notin \mathfrak {m}^{\lceil r\xi (\boldsymbol{\beta} )\rceil }$. Since 
$\mathfrak {a}_\bullet$ is linearly bounded by 
$\mathfrak {m}$, there exists 
$c>0$ such that 
$\mathfrak {a}_{ck}\subset \mathfrak {m}^k$ for any 
$k\in \mathbb {Z}_{>0}$. In particular, we get 
$f\notin \mathfrak {a}_{c\lceil r\xi (\boldsymbol{\beta} )\rceil }$. So 
$m(\boldsymbol{\beta} )\le M_2\xi (\boldsymbol{\beta} )$, where 
$M_2:= cr$.
(3) Assume that 
$\mathfrak {a}_\bullet =\mathfrak {a}_\bullet (u)$. Then for any 
$k\in \mathbb {N}$, we have 
$f^k \in v^{-1}(kv(f))\cap \mathfrak {a}_{ku(f)}(u)$ which implies that
\[ m(kv(f)) \geq ku(f). \]
Note that 
$\lim _{k\to \infty } ({m(kv(f))}/{k})\le M_2$ exists by (1) and (2). Letting 
$k\to \infty$, we get the inequality. If 
$u=w$, then by definition we have 
$m(\boldsymbol{\beta} )=\langle \boldsymbol{\alpha},\boldsymbol{\beta} \rangle$, which implies the equality immediately.
(4) Choose finitely generated subsemigroups 
$\mathcal {S}_1\subset \mathcal {S}_2\subset \cdots$ of 
$\mathcal {S}$ such that 
$\bigcup _{l=1}^\infty \mathcal {S}_l=\mathcal {S}$ and each 
$\mathcal {S}_l$ generates 
$\mathbb {Z}^n$ as a group. By [Reference KhovanskiiKho92, §3, Proposition 3], there exists 
$\gamma _l\in \mathcal {S}_l$ such that
\[ (C(\mathcal{S}_l)+\gamma_l)\cap \mathbb{N}^n\subset \mathcal{S}_l. \]
For any 
$\boldsymbol{\beta} \in \mathrm {Int}(C(\mathcal {S}_l))$, there exists 
$k_0$ such that for any 
$k\ge k_0$, we have
\[ \lfloor k\boldsymbol{\beta} \rfloor \in(C(\mathcal{S}_l)+\gamma_l)\cap \mathbb{N}^n\subset \mathcal{S}_l. \]
So we can define
\begin{equation} h(\boldsymbol{\beta}):= \limsup_{k\to\infty}\frac{m(\lfloor k\boldsymbol{\beta} \rfloor)}{k}. \end{equation} We first show that this is indeed a limit. Since 
$\boldsymbol{\beta} \in \mathrm {Int}(C(\mathcal {S}_l))$, there exists 
$d\in \mathbb {Z}_{>0}$ such that 
$B(d\boldsymbol{\beta},3)\subset C(\mathcal {S}_l)+\gamma _l$, where we use the maximum norm on 
$\mathbb {R}^n$. Hence, for any 
$k_1,k_2\ge k_0$, we have
\[ \lfloor (k_1+k_2+d)\boldsymbol{\beta} \rfloor -\big(\lfloor k_1\boldsymbol{\beta} \rfloor + \lfloor k_2\boldsymbol{\beta} \rfloor\big)\in C(\mathcal{S}_l)+\gamma_l\subset \mathcal{S}_l. \]
By (1) we have
\[ m\big(\lfloor (k_1+k_2+d)\boldsymbol{\beta} \rfloor\big)\ge m\big(\lfloor k_1\boldsymbol{\beta} \rfloor\big)+m\big(\lfloor k_2\boldsymbol{\beta} \rfloor\big). \]
This shows that the function 
$k\mapsto m(\lfloor (k-d)\boldsymbol{\beta} \rfloor )$ is sup-additive for 
$k\ge k_0+d$. So by (2) and Fekete's lemma we know that there exists
\[ \lim_{k\to\infty} \frac{m(\lfloor k\boldsymbol{\beta} \rfloor)}{k}=\lim_{k\to\infty} \frac{m(\lfloor (k-d)\boldsymbol{\beta} \rfloor)}{k}\le M_2. \]
Since the above definition does not depend on 
$l$, we have a well-defined function 
$h$ on 
$\mathrm {Int}(C(\mathcal {S}))=\cup _{l=1}^\infty \mathrm {Int}(C(\mathcal {S}_l))$. Clearly 
$h$ is homogeneous of degree 
$1$.
Next we prove the continuity of 
$h$. It suffices to prove the continuity of 
$h$ in 
$\mathrm {Int}(C(\mathcal {S}_l))$. We first show that for any 
$\boldsymbol{\beta},\boldsymbol{\beta} _1\in \mathrm {Int}(C(\mathcal {S}_l))$, we have
\begin{equation} h(\boldsymbol{\beta}+\boldsymbol{\beta}_1)\ge h(\boldsymbol{\beta}). \end{equation}
Indeed, there exists 
$k_0$ such that for any 
$k>k_0$, we have 
$B(k\boldsymbol{\beta},2)\subset C(\mathcal {S}_l)+\gamma _l$. Hence,
\[ \lfloor k(\boldsymbol{\beta}+\boldsymbol{\beta}_1)\rfloor -\lfloor k\boldsymbol{\beta} \rfloor\in \mathcal{S}_l, \]
and by (1) we get 
$m(\lfloor (k(\boldsymbol{\beta} +\boldsymbol{\beta} _1 \rfloor ))\ge m(\lfloor k\boldsymbol{\beta} \rfloor ))$. Dividing by 
$k$ and letting 
$k\to \infty$, we get (A.4). Now for any 
$\epsilon \in (0,1)$ and 
$\boldsymbol{\beta} \in \mathrm {Int}(C(\mathcal {S}_l))$, there exists 
$\rho >0$ such that 
$B(\epsilon \boldsymbol{\beta},\rho )\subset \mathrm {Int}(C(\mathcal {S}_l))$. Hence, for any 
$\boldsymbol{\beta} '\in B(\boldsymbol{\beta},\rho )$, we have 
$\epsilon \boldsymbol{\beta} \pm (\boldsymbol{\beta} -\boldsymbol{\beta} ')\in \mathrm {Int}(C(\mathcal {S}_l))$. By (A.4) and the homogeneity of 
$h$, we get
\[ (1-\epsilon)h(\boldsymbol{\beta})=h((1-\epsilon)\boldsymbol{\beta})\le h(\boldsymbol{\beta}')\le h((1+\epsilon)\boldsymbol{\beta})=(1+\epsilon)h(\boldsymbol{\beta}). \]
Hence, 
$h$ is continuous on 
$\mathrm {Int}(C(\mathcal {S}_l))$.
The concavity of 
$h$ follows easily from its continuity, homogeneity, and (1).
(5) By definition we have
\[ m_{\mathfrak{a}_\bullet}(\boldsymbol{\beta})= \lambda\cdot m_{\mathfrak{a}_{\lambda\bullet}}(\boldsymbol{\beta}) \]
for any 
$\boldsymbol{\beta} \in \mathcal {S}$. So we have
\[ h_{\mathfrak{a}_\bullet}(\boldsymbol{\beta})=\lim_{m\to\infty}\frac{m_{\mathfrak{a}_\bullet}(k\boldsymbol{\beta})}{k}=\lambda\cdot h_{\mathfrak{a}_{\lambda\bullet}}(\boldsymbol{\beta}) \]
for any 
$\boldsymbol{\beta} \in \mathcal {S}$. Now (A.2) follows easily from the homogeneity and continuity of 
$h$.
The following proposition asserts that 
$f_{\boldsymbol{\beta} }$ gives a basis for 
$\mathfrak {a}_m/\mathfrak {a}_{>m}$.
Proposition A.6 For any 
$m\in \mathbb {R}_{\geq 0}$, the set 
$\{[f_{\boldsymbol{\beta} }]\mid m(\boldsymbol{\beta} )=m\}$ forms a basis for the 
$\mathbb {k}$-vector space 
$\mathfrak {a}_m/\mathfrak {a}_{>m}$.
Proof. We may assume that 
$m$ is a jumping number of 
$\mathfrak {a}_\bullet$. Let
\[ \mathfrak{B}_m:= \{\boldsymbol{\beta}\in\mathcal{S}\mid m(\boldsymbol{\beta})=m\}. \]
We first show that the set 
$\{[f_{\boldsymbol{\beta} }]\}$ is linearly independent. Assume to the contrary that there exist 
$c_{\boldsymbol{\beta} }\in \mathbb {k}$, not all zero, such that
\[ g:= \sum_{\boldsymbol{\beta}\in\mathfrak{B}_m} c_{\boldsymbol{\beta}}f_{\boldsymbol{\beta}}\in \mathfrak{a}_{>m}. \]
Then we have 
$v(g)=\boldsymbol{\beta} _0:= \min \{\boldsymbol{\beta} \in \mathfrak {B}_m\mid c_{\boldsymbol{\beta} }\ne 0\}$, which implies 
$m(\boldsymbol{\beta} _0)>m$, a contradiction.
It remains to prove that for any 
$f\in \mathfrak {a}_m\backslash \mathfrak {a}_{>m}$, there exist 
$c_{\boldsymbol{\beta} }\in \mathbb {k}$ for each 
$\boldsymbol{\beta} \in \mathfrak {B}_m$ such that
\begin{equation} f-\sum_{\boldsymbol{\beta}\in\mathfrak{B}_m} c_{\boldsymbol{\beta}}f_{\boldsymbol{\beta}}\in \mathfrak{a}_{>m}. \end{equation}
By condition (1) of Definition A.1, there exists a unique sequence 
$\{a_{\boldsymbol{\beta} }\}_{\boldsymbol{\beta} \in \mathcal {S}}$ such that for any 
$\boldsymbol{\beta} '\in \mathcal {S}$, we have
\[ w\biggl(f-\sum_{\boldsymbol{\beta}\preceq \boldsymbol{\beta}'}a_{\boldsymbol{\beta}}f_{\boldsymbol{\beta}}\biggr)>\langle \boldsymbol{\alpha},\boldsymbol{\beta}' \rangle. \]
Let 
$\boldsymbol{\beta} _0$ be as defined in Lemma A.4. In particular, we have 
$\boldsymbol{\beta} \prec \boldsymbol{\beta} _0$ for any 
$\boldsymbol{\beta} \in \mathfrak {B}_m$. Write
\[ f-\sum_{\boldsymbol{\beta}\in\mathfrak{B}_m} a_{\boldsymbol{\beta}}f_{\boldsymbol{\beta}}=\biggl(f-\sum_{\boldsymbol{\beta}\prec \boldsymbol{\beta}_0} a_{\boldsymbol{\beta}}f_{\boldsymbol{\beta}}\biggr) +\sum_{\boldsymbol{\beta}\notin\mathfrak{B}_m,\boldsymbol{\beta}\prec \boldsymbol{\beta}_0}a_{\boldsymbol{\beta}}f_{\boldsymbol{\beta}}=:g_1+g_2. \]
By the choice of 
$a_{\boldsymbol{\beta} }$, we know that 
$v(g_1)\ge \boldsymbol{\beta} _0$. Hence, 
$g_1\in \mathfrak {a}_{>m}$ by the choice of 
$\boldsymbol{\beta} _0$.
Claim For any 
$\boldsymbol{\beta} \preceq \boldsymbol{\beta} _0$ with 
$\boldsymbol{\beta} \notin \mathfrak {B}_m$, if 
$a_{\boldsymbol{\beta} }\ne 0$, then 
$m(\boldsymbol{\beta} )> m$.
We prove the claim by induction. Let 
$\boldsymbol{\beta} \preceq \boldsymbol{\beta} _0$ and assume that the claim is true for any 
$\boldsymbol{\beta} '\prec \boldsymbol{\beta}$. Then 
$f_{\boldsymbol{\beta} '}\in \mathfrak {a}_m$ for 
$\boldsymbol{\beta} '\prec \boldsymbol{\beta}$, and hence 
$f-\sum _{\boldsymbol{\beta} '\prec \boldsymbol{\beta} }a_{\boldsymbol{\beta} '}f_{\boldsymbol{\beta} '}\in \mathfrak {a}_m$. Now 
$a_{\boldsymbol{\beta} }\ne 0$, so we have
\[ v\biggl(f-\sum_{\boldsymbol{\beta}'\prec \boldsymbol{\beta}}a_{\boldsymbol{\beta}'}f_{\boldsymbol{\beta}'}\biggr)=v(a_{\boldsymbol{\beta}}f_{\boldsymbol{\beta}})=\boldsymbol{\beta}. \]
Hence, 
$v^{-1}(\boldsymbol{\beta} )\cap \mathfrak {a}_m\ne \emptyset$ and 
$m(\boldsymbol{\beta} )\ge m$. Since 
$m(\boldsymbol{\beta} )\ne m$ by assumption, we have strict inequality and the claim is proved.
By the claim we know that 
$g_2\in \mathfrak {a}_{>m}$. Thus we conclude that
\[ f-\sum_{\boldsymbol{\beta}\in\mathfrak{B}_m} a_{\boldsymbol{\beta}}f_{\boldsymbol{\beta}}\in\mathfrak{a}_{>m}, \]
that is, (A.5) holds with 
$c_{\boldsymbol{\beta} }=a_{\boldsymbol{\beta} }$.
A.4 Multiplicities and Okunkov bodies
In this section we apply the strategy as in [Reference Lazarsfeld and MustaLM09, Reference Kaveh and KhovanskiiKK14] to estimate the multiplicities.
Let 
$\mathfrak {a}_{\bullet,i}$ be two linearly bounded 
$\mathfrak {m}$-filtrations for 
$i=0,1$. Recall that the geodesic 
$(\mathfrak {a}_{\bullet,t})_{t\in [0,1]}$ between 
$\mathfrak {a}_{\bullet,0}$ and 
$\mathfrak {a}_{\bullet,1}$ is defined as
\[ \mathfrak{a}_{\lambda,t}:= \sum_{\lambda=(1-t)\mu+ t\nu} (\mathfrak{a}_{\mu,0}\cap \mathfrak{a}_{\nu,1}). \]
Then 
$\mathfrak {a}_{\bullet,t}$ is also a linearly bounded 
$\mathfrak {m}$-filtration.
Theorem A.7 Let 
$w\in \mathrm {Val}_{X,x}$ be a quasi-monomial valuation. Denote 
$\mathfrak {a}_{\bullet,0}:=\mathfrak {a}_{\bullet }(w)$. Let 
$\mathfrak {a}_{\bullet,1}$ be a linearly bounded 
$\mathfrak {m}$-filtration. Let 
$(\mathfrak {a}_{\bullet,t})_{t\in [0,1]}$ be the geodesic between 
$\mathfrak {a}_{\bullet,0}$ and 
$\mathfrak {a}_{\bullet,1}$. The function 
$E(t) \colon [0,1]\to \mathbb {R}$ defined by 
$E(t) := \mathrm {e}(\mathfrak {a}_{\bullet,t})$ satisfies the following properties.
(1)
$E(t)$ is smooth;(2)
$E(t)^{-1/n}$ is concave, meaning
\[ E(t)^{-1/n}\ge (1-t)E(0)^{-1/n}+tE(1)^{-1/n} \quad \text{for all } t\in [0,1]. \]
(3) Suppose, in addition,
$\mathfrak {a}_{\bullet,1} = \mathfrak {a}_{\bullet }(w')$ for some valuation $w'\in \mathrm {Val}_{X,x}$. Then $E(t)^{-1/n}$ is linear if and only if $w' = cw$ for some $c\in \mathbb {R}_{>0}$.
From now on, we follow the notation of Theorem A.7. Let 
$v$ be a good valuation associated to 
$w$ given by Lemma A.2.
Lemma A.8 Let 
$\mathfrak {a}_{\bullet,t}$ be as above for 
$t\in [0,1]$. Let 
$m^{(t)}=m_{\mathfrak {a}_{\bullet,t}}:\mathcal {S}\to \mathbb {R}_{\geq 0}$ be the function defined in (A.1). Then we have
\[ m^{(t)}(\boldsymbol{\beta})= (1-t)m^{(0)}(\boldsymbol{\beta})+tm^{(1)}(\boldsymbol{\beta}) \]
for any 
$t\in [0,1]$ and any 
$\boldsymbol{\beta} \in \mathcal {S}$.
Proof. Fix an arbitrary element 
$\boldsymbol{\beta} \in \mathcal {S}$. For simplicity, denote 
$m_t:=m^{(t)}(\boldsymbol{\beta} )$ for 
$t\in [0,1]$.
We first show that 
$m_t\geq (1-t)m_0 + tm_1$. By assumption, we have 
$w(f) = \langle \boldsymbol{\alpha}, v(f)\rangle$ for any 
$f\in R\setminus \{0\}$. Thus
\[ m_0 =m^{(0)}(\boldsymbol{\beta})= \max\{w(f)\mid f\in R\setminus\{0\} \textrm{ and }v(f)=\boldsymbol{\beta}\} = \langle \boldsymbol{\alpha}, \boldsymbol{\beta}\rangle. \]
Pick 
$g\in v^{-1}(\boldsymbol{\beta} )\cap \mathfrak {a}_{m_1,1}$. Then clearly 
$w(g) = \langle \boldsymbol{\alpha}, v(g)\rangle = \langle \boldsymbol{\alpha}, \boldsymbol{\beta} \rangle = m_0$. Thus 
$g\in \mathfrak {a}_{m_0, 0}$ which implies that 
$v^{-1}(\boldsymbol{\beta} )\cap \mathfrak {a}_{m_1,1} \cap \mathfrak {a}_{m_0,0}$ is non-empty. Since 
$\mathfrak {a}_{m_1,1} \cap \mathfrak {a}_{m_0,0}\subset \mathfrak {a}_{(1-t)m_0+tm_1,t}$, we know that 
$v^{-1}(\boldsymbol{\beta} )\cap \mathfrak {a}_{(1-t)m_0+tm_1,t}\neq \emptyset$ which implies 
$m_t\geq (1-t)m_0+tm_1$ by (A.1).
Next, we show that 
$m_t\leq (1-t)m_0+tm_1$. Choose 
$f\in v^{-1}(\boldsymbol{\beta} ) \cap \mathfrak {a}_{m_t, t}$. We may write 
$f=\sum _{i} f_i$ as a finite sum such that 
$f_i \in \mathfrak {a}_{\mu _i, 0}\cap \mathfrak {a}_{\nu _i,1}$, where 
$(1-t)\mu _i +t\nu _i=m_t$ for every 
$i$. Since 
$f_i\in \mathfrak {a}_{\mu _i,0}=\mathfrak {a}_{\mu _i}(w)$, we have 
$w(f_i)\geq \mu _i$. Hence, after replacing 
$(\mu _i, \nu _i)$ by 
$(w(f_i), t^{-1}(m_t-(1-t)w(f_i)))$ the assumption 
$f_i \in \mathfrak {a}_{\mu _i, 0}\cap \mathfrak {a}_{\nu _i,1}$ still holds. Furthermore, if we have 
$\mu _i=\mu _j$ for some 
$i\neq j$ then we may replace 
$(f_i, f_j)$ by 
$f_i+f_j$. After finitely many steps of replacements and permutations, we obtain a decomposition 
$f=\sum _{i=1}^{l} f_i$ such that 
$f_i\in \mathfrak {a}_{\mu _i,0}\cap \mathfrak {a}_{\nu _i,1}$ where 
$(1-t)\mu _i + t\nu _i = m_t$ and 
$\mu _i=w(f_i)$ for every 
$i$, and 
$\mu _1<\mu _2<\cdots <\mu _l$. Denote 
$\boldsymbol{\beta} _i:= v(f_i)$. Since 
$\mu _i = \langle \boldsymbol{\alpha}, \boldsymbol{\beta} _i\rangle$, we know that 
$\boldsymbol{\beta} _1\prec \boldsymbol{\beta} _2\prec \cdots \prec \boldsymbol{\beta} _l$. As a result, we have
\[ \boldsymbol{\beta} = v(f) = v\biggl(\sum_{i=1}^l f_i\biggr) = v(f_1)= \boldsymbol{\beta}_1. \]
Thus, 
$f_1\in v^{-1}(\boldsymbol{\beta} )\cap \mathfrak {a}_{\nu _1,1}$, which implies that 
$m_1\geq \nu _1$ by (A.1). Meanwhile, we have
\[ \mu_1=w(f_1)=\langle\boldsymbol{\alpha},\boldsymbol{\beta}\rangle = m_0. \]
Therefore,
\[ m_t= (1-t)\mu_1 + t\nu_1\leq (1-t)m_0 + tm_1 , \]
which completes the proof.
As an immediate corollary to Lemma A.8, we get the following result.
Corollary A.9 Let 
$h_t:= h_{\mathfrak {a}_{\bullet,t}}:\mathrm {Int}(C(\mathcal {S}))\to \mathbb {R}_{>0}$ be defined as in (A.3) for 
$t\in [0,1]$. Then we have 
$h_t=(1-t)h_0+th_1$.
The following lemma can be viewed as an Izumi-type estimate in our setting of filtrations.
Lemma A.10 There exists 
$M>0$ depending only on 
$w$ and 
$\mathfrak {a}_{\bullet,1}$ such that for any 
$f\in R\backslash \{0\}$ with 
$\xi (v(f))\ge M$ and 
$t\in [0,1]$, we have
\[ f\in \mathfrak{a}_{M^{-1}\cdot \xi(v(f)),t}. \]
Proof. Recall that we may define 
$\xi :\mathbb {R}^n\to \mathbb {R}$ in Definition A.1 by 
$\xi (v(f))=w(f)$. Moreover, there exists 
$r>0$ such that 
$\mathrm {ord}_\mathfrak {m}(f)\ge r\xi (v(f))$. Take 
$d\in \mathbb {Z}_{>0}$ such that 
$\mathfrak {m}^d\subset \mathfrak {a}_{1,i}$ for 
$i=0,1$. Set 
$M':= 2d\cdot r^{-1}$. For any 
$f\in R$ with 
$\xi (v(f))\ge M'$, we have 
$\mathrm {ord}_\mathfrak {m}(f)\ge r\cdot \xi (v(f))\ge 2d$. Hence,
\[ f\in \mathfrak{m}^{\lceil r\xi(v(f)) \rceil}\subset \mathfrak{a}_{\lfloor {r\xi(v(f))}/{d}\rfloor,i}\subset \mathfrak{a}_{M'^{-1}\cdot \xi(v(f)),i}, \]
for 
$i=0,1$, where we used the inequality 
$\lfloor \lambda \rfloor \ge \lambda /2$ for 
$\lambda \ge 2$. Hence, when 
$\lambda := M'^{-1}\cdot \xi (v(f))\ge 2$, we have
\[ f\in \mathfrak{a}_{\lambda,0}\cap \mathfrak{a}_{\lambda,1} \subset \mathfrak{a}_{\lambda,t}, \]
and the lemma is proved with 
$M=2M'$.
In view of the above lemma, we define
\[ \Gamma:= \{(\boldsymbol{\beta},m)\in\mathcal{S}\times \mathbb{N}\mid \xi(\boldsymbol{\beta})\le Mm\} \quad \text{ and }\quad \Gamma_m:= \Gamma \cap (\mathcal{S}\times \{m\}) . \]
For 
$\boldsymbol{\beta} \in \mathcal {S}$ and 
$t\in [0,1]$, recall that 
$m^{(t)}(\boldsymbol{\beta} ) := m_{\mathfrak {a}_{m,t}}(\boldsymbol{\beta} )$ and choose an element 
$f^{(t)}_{\boldsymbol{\beta} }:= f_{\mathfrak {a}_{m,t},\boldsymbol{\beta} }\in v^{-1}(\boldsymbol{\beta} )\cap \mathfrak {a}_{m^{(t)}(\boldsymbol{\beta} ),t}$. Then we can define
\[ \Gamma^{(t)}:= \{(\boldsymbol{\beta},m)\in\Gamma\mid f^{(t)}_{\boldsymbol{\beta}}\in\mathfrak{a}_{m,t}\} \quad\text{and}\quad \Gamma^{(t)}_m := \Gamma^{(t)} \cap (\mathcal{S}\times \{m\}). \]
We will use these sets to estimate the multiplicities of 
$\mathfrak {a}_{\bullet,t}$. The following property for 
$\Gamma ^{(t)}$ is one of the main ingredients for our proof of Theorem A.7.
Proposition A.11 Let 
$m\in \mathbb {N}$. Then
\[ \ell(R/\mathfrak{a}_{m,t})=\#(\Gamma_m\backslash\Gamma^{(t)}_m) \]
for any 
$t\in [0,1]$.
Proof. Note that the jumping numbers of the 
$\mathfrak {m}$-filtration 
$\mathfrak {a}_{\bullet,t}$ form a discrete set as for any 
$\mu >\lambda >0$, 
$\ell (\mathfrak {a}_{\lambda,t}/\mathfrak {a}_{\mu,t})<\infty$. By Proposition A.6, the quotient ring 
$R/\mathfrak {a}_{m,t}$ has a basis 
$\{[f^{(t)}_{\boldsymbol{\beta} }]\mid f^{(t)}_{\boldsymbol{\beta} }\notin \mathfrak {a}_{m,t}\}$. Note that if 
$\xi (v(f^{(t)}_{\boldsymbol{\beta} }))\ge Mm$ for some 
$\boldsymbol{\beta} \in \mathcal {S}$, then 
$f^{(t)}_{\boldsymbol{\beta} }\in \mathfrak {a}_{M^{-1}\cdot \xi (v(f^{(t)}_{\boldsymbol{\beta} })),t}\subset \mathfrak {a}_{m,t}$ by Lemma A.10. Hence, for any 
$\boldsymbol{\beta}$ satisfying 
$f^{(t)}_{\boldsymbol{\beta} }\notin \mathfrak {a}_{m,t}$, we have 
$\xi (\boldsymbol{\beta} )=\xi (v(f^{(t)}_{\boldsymbol{\beta} }))\le Mm$, that is, 
$\boldsymbol{\beta} \in \Gamma _m$. Thus
\[ \Gamma_m\backslash\Gamma^{(t)}_m=\{\boldsymbol{\beta}\mid f^{(t)}_{\boldsymbol{\beta}}\notin \mathfrak{a}_{m,t}\}, \]
and the proposition follows.
We now check that the 
$\Gamma$ and 
$\Gamma ^{(t)}$ are semigroups satisfying conditions (2.3-5) of [Reference Lazarsfeld and MustaLM09].
Lemma A.12 Let 
$t\in [0,1]$. Then
(1)
$\Gamma$ and $\Gamma ^{(t)}$ are semigroups,(2)
$\Gamma _0=\Gamma ^{(t)}_0=\{0\}$,(3) there exist finitely many vectors
$(v_j,1)$ spanning a semigroup $B\subset \mathbb {N}^{n+1}$ such that
and\[ \Gamma^{(t)}\subset \Gamma\subset B, \]
(4) if we replace
$M$ by a proper multiple of it in the definition of $\Gamma$, then $\Gamma ^{(t)}$ and $\Gamma$ contain a set of generators of $\mathbb {Z}^{n+1}$ as a group.
Proof. (1) This follows from the sup-additivity of 
$m(\cdot )$, Proposition A.5(1).
(2) This is clear.
(3) Choose 
$b\in \mathbb {N}$ such that 
$b\cdot \min \{\alpha _i\mid 1\le i\le r\}>M$. Then it is easy to see that 
$\Gamma$ is contained in the semigroup 
$B$ generated by 
$\{(\beta _1,\ldots,\beta _n,1)\mid 0\le \beta _i\le b\}$.
(4) It suffices to show that 
$\Gamma ^{(t)}$ generates 
$\mathbb {Z}^{n+1}$. To this end, recall that as in the proof of Lemma A.2, 
$\mathcal {S}$ contains some 
$\boldsymbol{\beta} '$ and 
$\boldsymbol{\beta} '+\boldsymbol{e}_i$ for 
$1\le i\le n$. We may assume that 
$\xi (\boldsymbol{\beta} ')\ge 2M$. Let 
$m:= \lfloor M^{-1}\cdot \xi (\boldsymbol{\beta} ') \rfloor -1>0$. Then by Lemma A.10 we have
\[ f_{\boldsymbol{\beta}'}\in\mathfrak{a}_{M^{-1}\cdot\xi(\boldsymbol{\beta}'),t}\subset \mathfrak{a}_{m+1,t}\subset \mathfrak{a}_{m,t} \]
and
\[ f_{\boldsymbol{\beta}'+\boldsymbol{e}_i}\in\mathfrak{a}_{M^{-1}\cdot\xi(\boldsymbol{\beta}'+\boldsymbol{e}_i),t}\subset \mathfrak{a}_{m,t}. \]
If we replace 
$M$ by 
$M'=2M$ in the definition of 
$\Gamma$, then
\[ \xi(\boldsymbol{\beta}')=M'\cdot \frac{M^{-1}\cdot\xi(\boldsymbol{\beta}')}{2}\le M'\cdot \lfloor M^{-1}\cdot\xi(\boldsymbol{\beta}') \rfloor =M'm, \]
where we used the inequality 
$\lfloor \lambda \rfloor \ge \lambda /2$ for 
$\lambda \ge 2$. Now by definition, 
$\Gamma ^{(t)}$ contains the vectors 
$(\boldsymbol{\beta} ',m)$, 
$(\boldsymbol{\beta} '+\boldsymbol{e}_i,m)$, 
$1\le i\le n$ and 
$(\boldsymbol{\beta} ',m+1)$, which implies that 
$\Gamma ^{(t)}$ generates 
$\mathbb {Z}^{n+1}$ as a group.
From now on we assume that 
$\Gamma$ and 
$\Gamma ^{(t)}$ satisfy the conditions of Lemma A.12. As in [Reference Lazarsfeld and MustaLM09, (2.1)], we define 
$\Delta$ as the closed convex hull of 
$\bigcup _m \Gamma _m/m$, and define 
$\Delta ^{(t)}$ as the closed convex hull of 
$\bigcup _m \Gamma _m^{(t)}/m$ for 
$t\in [0,1]$. Then 
$\Delta ^{(t)}\subset \Delta$. By Lemma A.12, Proposition A.11 and [Reference Lazarsfeld and MustaLM09, Proposition 2.1], we have the following result.
Corollary A.13 With notation as above,
\[ \mathrm{e}(\mathfrak{a}_{\bullet,t})=n!(\mathrm{vol}(\Delta)-\mathrm{vol}(\Delta^{(t)})) \]
for 
$t\in [0,1]$.
Next, we show that the 
$\Gamma ^{(t)}$ can be characterized by the function 
$h_t$ defined as in Corollary A.9.
Proposition A.14 Let 
$h_t:= h_{\mathfrak {a}_{\bullet,t}}$ for 
$t\in [0,1]$, where 
$h$ is defined by (A.3). Then
(1)
$\mathrm {Int}(\Delta ^{(t)})=\mathrm {Int}(\Delta )\cap (h_t)_{>1}$, and(2)
$\mathrm {Int}(\Delta )\backslash \Delta ^{(t)}=\mathrm {Int}(C(\mathcal {S}))\cap (h_t)_{<1}$.
Proof. (1) First we show that 
$\mathrm {Int}(\Delta ^{(t)})\subset \mathrm {Int}(\Delta )\cap (h_t)_{>1}$. It suffices to show that for any 
$\boldsymbol{\beta} \in \Delta ^{(t)}\cap \mathrm {Int}(C(\mathcal {S}))$ we have 
$h_t(\boldsymbol{\beta} )\ge 1$. Recall that 
$\Delta ^{(t)}$ is the closed convex hull of 
$\cup _m \Gamma ^{(t)}_m/m$. Hence, by the concavity of 
$h_i$ (Proposition A.5(4)), it suffices to show that for any 
$(\boldsymbol{\beta},m)\in \Gamma ^{(t)}\cap \mathrm {Int}(C(\mathcal {S}))$ we have 
$h_t(\boldsymbol{\beta} )\ge m$. But this follows immediately from (A.3) and the definition of 
$\Gamma ^{(t)}$.
To prove the converse, assume that 
$\boldsymbol{\beta} \in \mathrm {Int}(\Delta )\cap (h_t)_{>1}$. Let 
$\epsilon := ({h_t(\boldsymbol{\beta} )-1})/{3}>0$. Applying the proof of Proposition A.5(4), we know that there exists 
$m_1$ such that for any 
$m\ge m_1$, there exists 
$\boldsymbol{\beta} _m\in \Gamma _m$ with 
$\lim _{m} ({\boldsymbol{\beta} _m}/{m})=\boldsymbol{\beta}$. Since 
$h_t$ is continuous, we may choose 
$m_2\ge m_1$ such that for any 
$m\ge m_2$, we have 
$h_t({\boldsymbol{\beta} _m}/{m})>1+2\epsilon$. By (A.3), for each 
$m\ge m_2$, we may choose 
$k_m$ such that 
$(1+\epsilon )m^{(i)}(k_m\boldsymbol{\beta} _m)\ge h_i(\boldsymbol{\beta} _m)$ for 
$i=0,1$. Then for 
$m\ge m_2$, we have
\[ m^{(t)}(k_m\boldsymbol{\beta}_m)\ge \frac{1}{1+\epsilon}h_t(k_m\boldsymbol{\beta}_m) \ge \frac{1+2\epsilon}{1+\epsilon}k_mm> k_mm. \]
This shows that 
$k_m\boldsymbol{\beta} _m\in \Gamma ^{(t)}_{k_mm}$, hence
\[ \boldsymbol{\beta}=\lim_{m\to\infty} \frac{k_m\boldsymbol{\beta}_m}{k_mm}\in \Delta^{(t)}. \]
(2) By (1) we have 
$\mathrm {Int}(\Delta )\backslash \Delta ^{(t)}=\mathrm {Int}(\Delta )\cap (h_t)_{<1}$, so it suffices to show that for any 
$\boldsymbol{\beta} \in \mathrm {Int}(C(\mathcal {S}))\backslash \Delta$, we have 
$h_t(\boldsymbol{\beta} )>1$. By definition we have
\[ \Delta=C(\Gamma)\cap (\mathbb{R}^n\times\{1\})=C(\mathcal{S})\cap \xi_{\le M}. \]
Hence, by Proposition A.5 and Lemma A.10, we have
\begin{align*} h_t(\boldsymbol{\beta})&=\lim_k \frac{m^{(t)}(\lfloor k\boldsymbol{\beta} \rfloor)}{k}\ge \lim_k\frac{M^{-1}\cdot \xi(\lfloor k\boldsymbol{\beta} \rfloor)}{k}\\ &=M^{-1}\cdot\xi(\boldsymbol{\beta})>1, \end{align*}
where in the first inequality and the second equality we used the fact that for any 
$k\in \mathbb {Z}_{>0}$,
\[ k\xi(\boldsymbol{\beta})\ge \xi(\lfloor k\boldsymbol{\beta} \rfloor)=\langle \boldsymbol{\alpha},\lfloor k\boldsymbol{\beta} \rfloor \rangle\ge k\xi(\boldsymbol{\beta})-\|\boldsymbol{\alpha}\|_{1}. \]
The proof is finished.
Next we prove Theorem A.7.
Proof of Theorem A.7 By Lemma A.3, Corollary A.13 and Proposition A.14, we have
\begin{equation} E(t)=n!\,\mathrm{vol}(\Delta\backslash\Delta^{(t)})=n!\,\mathrm{vol}(\mathrm{Int}(C(\mathcal{S}))\cap (h_t)_{<1})=\int_{C(\mathcal{S})} e^{-h_t}\,d\mu. \end{equation}
By Corollary A.9 we have 
$h_t=(1-t)h_0+th_1$, hence 
$({d}/{dt})h_t=h_1-h_0$. By Proposition A.5(4), we can differentiate under the integral sign, and hence 
$E(t)$ is a smooth function. Thus (1) is proved.
To prove (2), we use a slight variant of the formula (A.6). Let 
$h:= h_1-h_0$. Then by homogeneity we have
\begin{align*} E''(t)&=\int_{C(\mathcal{S})} h^2e^{-h_t}\,d\mu\\ &=\int_0^\infty \,d\lambda\int_{C(\mathcal{S})\cap (h_t)_{=\lambda}}h^2e^{-h_t}\,d\mu_{n-1}\\ &=\int_0^\infty \lambda^{n+1}e^{-\lambda}\,d\lambda \int_{C(\mathcal{S})\cap (h_t)_{=1}}h^2\,d\mu_{n-1}\\ &=(n+1)!\int_{C(\mathcal{S})\cap (h_t)_{=1}}h^2\,d\mu_{n-1}, \end{align*}
where 
$\mu _{n-1}$ is the Lebesgue measure on 
$\mathbb {R}^{n-1}$. Similarly, we have
\[ E'(t)=n!\int_{C(\mathcal{S})\cap (h_t)_{=1}}h\,d\mu_{n-1} \]
and
\[ E(t)=(n-1)!\int_{C(\mathcal{S})\cap (h_t)_{=1}}\,d\mu_{n-1}. \]
Hence, by Cauchy–Schwarz we have
\begin{equation} E''(t)E(t)\ge \frac{n+1}{n}E'(t)^2\ge E'(t)^2, \end{equation}which implies (2).
To prove (3), first assume that the equality in (2) holds. By (A.7) and Cauchy–Schwarz we know that 
$h_1-h_0=C_t$ is a constant on 
$(h_t)_{=1}$. In particular, 
$h_1-h_0=C_0$ on 
$(h_0)_{=1}$. Since 
$h_1$ and 
$h_0$ are 
$1$-homogeneous and positive on 
$C(\mathcal {S})$, we know that 
$h_0=ch_1$, where 
$c:=1/(C_0+1) >0$. By Proposition A.5(5), we know that
\begin{equation} h_0=h_{\mathfrak{a}_{\bullet,0}}=h_{\mathfrak{a}_{c^{-1}\bullet,1}}. \end{equation}By Lemma A.3, Corollary A.13 and Proposition A.14 we have
\[ \mathrm{vol}(w)=\mathrm{e}(\mathfrak{a}_{\bullet,0})=\mathrm{e}(\mathfrak{a}_{c^{-1}\bullet,1})=\mathrm{vol}(cw'). \]
Moreover, for any 
$f\in R\backslash \{0\}$, let 
$\boldsymbol{\beta} :=v(f)$. Then by Proposition A.5(3) we have 
$h_0(\boldsymbol{\beta} )=w(f)$ and 
$h_{\mathfrak {a}_{c^{-1}\bullet,1}}(\boldsymbol{\beta} )\ge cw'(f)$. Combined with (A.8), we get 
$cw'(f)\le w(f)$. So by [Reference Li and XuLX20, Proposition 2.7], we conclude that 
$w=c w'$.
Conversely, if 
$w=cw'$, then for any 
$\lambda \in \mathbb {R}_{>0}$, by a direct calculation we have
\[ \mathfrak{a}_{\lambda,t}=\sum_{(1-t)\mu+t\nu=\lambda}\mathfrak{a}_{\mu}(cw')\cap\mathfrak{a}_{\nu}(w')=\mathfrak{a}_{f(t,\lambda)}(w'), \]
where 
$f(t)={\lambda }/({t+c(1-t)})$. So 
$E(t)=({1}/({t+c(1-t)}))^n\,\mathrm {vol}(w')$, and the proof is finished.
Open access
