2 Preliminaries

Throughout this section, we state some results on convex functions. For general references, we refer to[Reference Rockafellar19, Reference Rockafellar and Wets20, Reference Schneider21].

For $v\in {\mbox {Conv}({\mathbb R}^n; {\mathbb R})}$ , we write $\partial v(x)$ for the subdifferential of v at $x\in {\mathbb R}^n$ , which is the set

$$\begin{align*}\partial v(x)= \{y\in{\mathbb R}^n : v(z)\geq v(x)+\langle y,z-x\rangle \; \forall z\in{\mathbb R}^n\}.\end{align*}$$

The function v is differentiable at x if and only if $\partial v(x)$ contains only one element – namely, the gradient $\nabla v(x)$ .

The Monge–Ampère measure of v, which is a Radon measure on ${\mathbb R}^n$ , is defined as

$$\begin{align*}\textrm{MA}(v;B)= V_n\left(\bigcup_{b\in B}\partial v(b)\right)\end{align*}$$

for Borel sets $B\subseteq {\mathbb R}^n$ (see, for example, [Reference Figalli10, Theorem 2.3]). The mixed Monge–Ampère measure, which is associated to an n-tuple of elements of ${\mbox {Conv}({\mathbb R}^n; {\mathbb R})}$ , is now given by the relation

(2.1) $$ \begin{align} \textrm{MA}(\lambda_1 v_1+\cdots + \lambda_m v_m;\cdot) = \sum_{i_1,\ldots,i_n=1}^m \lambda_{i_1}\cdots \lambda_{i_n} \textrm{MA}(v_{i_1},\ldots,v_{i_n};\cdot), \end{align} $$

where $m\in {\mathbb N}$ , $v_1,\ldots ,v_m\in {\mbox {Conv}({\mathbb R}^n; {\mathbb R})}$ , and $\lambda _1,\ldots ,\lambda _m\geq 0$ . Equation (2.1) uniquely determines the mixed Monge–Ampère measure if we additionally assume that it is symmetric in its entries. See [Reference Trudinger and Wang23] and [Reference Colesanti, Ludwig and Mussnig6, Theorem 4.3].

For a convex function $w\colon {\mathbb R}^n\to (-\infty ,\infty ]$ , we consider its convex conjugate or Legendre–Fenchel transform

$$\begin{align*}w^*(x)=\sup\left\{ \langle x,y\rangle - w(y): y\in{\mathbb R}^n \right\}\end{align*}$$

for $x\in {\mathbb R}^n$ . For each $v\in {\mbox {Conv}({\mathbb R}^n; {\mathbb R})}$ , the convex conjugate $v^*$ is a lower semicontinuous, convex function on ${\mathbb R}^n$ with values in $(-\infty ,\infty ]$ , which satisfies $v(\bar {x})<\infty $ for at least one $\bar {x}\in {\mathbb R}^n$ and which is super-coercive; that is,

$$\begin{align*}\lim_{|x|\to\infty} \frac{v^*(x)}{|x|}=\infty.\end{align*}$$

We denote the set of all such functions by ${\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ and remark this duality can be stated as $u^*\in {\mbox {Conv}({\mathbb R}^n; {\mathbb R})}$ if and only if $u\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ .

While the space of convex bodies is naturally embedded into ${\mbox {Conv}({\mathbb R}^n; {\mathbb R})}$ by associating with each body $K\in {\mathcal K}^n$ its support function $h_K\in {\mbox {Conv}({\mathbb R}^n; {\mathbb R})}$ , the canonical representative of K in ${\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ is given by its convex indicator function

(2.2) $$ \begin{align} \mathbf{I}_K(x)=h_K^*(x)=\begin{cases}0\quad&\text{if } x\in K,\\ \infty\quad&\text{else.} \end{cases} \end{align} $$

We equip the space ${\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ with the topology associated with epi-convergence, where a sequence of convex functions $w_j\colon {\mathbb R}^n\to (-\infty ,\infty ]$ , $j\in {\mathbb N}$ , epi-converges to $w\colon {\mathbb R}^n\to (-\infty ,\infty ]$ if for every $x\in {\mathbb R}^n$ ,

• • $w(x)\leq \liminf _{j\to \infty } w_j(x_j)$ for every sequence $x_j\to x$ and

• • $w(x)=\lim _{j\to \infty } w_j(x_j)$ for some sequence $x_j\to x$ .

By [Reference Rockafellar and Wets20, Theorem 11.34], convex conjugation is a homeomorphism between ${\mbox {Conv}({\mathbb R}^n; {\mathbb R})}$ and ${\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ . Let us remark that while on ${\mbox {Conv}({\mathbb R}^n; {\mathbb R})}$ epi-convergence coincides with pointwise convergence, this is not the case anymore on ${\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ . For an alternative description of epi-convergence on ${\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ which uses Hausdorff convergence of level sets, we refer to [Reference Colesanti, Ludwig and Mussnig4, Lemma 5].

We need the following result, which is a consequence of [Reference Colesanti, Ludwig and Mussnig9, Lemma 3.3].

Lemma 2.1 The map

$$\begin{align*}(\vartheta,u)\mapsto u\circ \vartheta\end{align*}$$

is jointly continuous on $\operatorname {SO}(n)\times {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ .

For a convex function $w\colon {\mathbb R}^n\to (-\infty ,\infty ]$ , we write

$$\begin{align*}\operatorname{epi}(w)=\{(x,t)\in{\mathbb R}^n\times{\mathbb R} : w(x)\leq t\}\end{align*}$$

for its epi-graph, which is a convex subset of ${\mathbb R}^n\times {\mathbb R}$ . For the convex conjugate of the pointwise sum of two functions $v_1$ and $v_2$ in ${\mbox {Conv}({\mathbb R}^n; {\mathbb R})}$ , we have

(2.3) $$ \begin{align} \operatorname{epi} (v_1+v_2)^*=\operatorname{epi}(v_1^*) + \operatorname{epi}(v_2^*). \end{align} $$

The corresponding operation on ${\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ is infimal convolution or epi-sum which for $u_1,u_2\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ is given by

$$\begin{align*}(u_1\mathbin{\Box} u_2) (x)=\inf\left\{u_1(x-y)+u_2(y):y\in{\mathbb R}^n \right\}\end{align*}$$

for $x\in {\mathbb R}^n$ . The set in (2.3) can now be written as $\operatorname {epi}(v_1^*\mathbin {\Box } v_2^*)$ . By the preceding exposition, the following result, which can be found in [Reference Rockafellar and Wets20, Theorem 7.46 (a)], is easy to see.

Next, for the convex conjugate of the pointwise multiplication of $v\in {\mbox {Conv}({\mathbb R}^n; {\mathbb R})}$ with $\lambda> 0$ , we have

where denotes the epi-multiplication of $u\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ with ${\lambda>0}$ . This operation continuously extends to $\lambda =0$ with .

The conjugate Monge–Ampère measure of $u\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ is defined by

$$\begin{align*}\textrm{MA}^*(u;\cdot)=\textrm{MA}(u^*;\cdot),\end{align*}$$

or equivalently,

$$\begin{align*}\int_{{\mathbb R}^n} \beta(y) \,\mathrm{d}\textrm{MA}^*(u;y)=\int_{\operatorname{dom}(u)} \beta(\nabla u(x))\,\mathrm{d} x\end{align*}$$

for measurable $\beta \colon {\mathbb R}^n\to [0,\infty )$ . Here,

$$\begin{align*}\operatorname{dom}(u)=\{x\in{\mathbb R}^n : u(x)<\infty\}\end{align*}$$

is the domain of u, and it follows from Rademacher’s theorem that a convex function is differentiable almost everywhere (w.r.t. the Lebesgue measure) on its domain. Similarly, for $u_1,\ldots ,u_n\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ , the conjugate mixed Monge–Ampère measure is given by

(2.4) $$ \begin{align} \textrm{MA}^{\!*}(u_1,\ldots,u_n;\cdot)=\textrm{MA}(u_1^*,\ldots,u_n^*;\cdot) \end{align} $$

and satisfies

(2.5)

for $m\in {\mathbb N}$ , $u_1,\ldots ,u_m\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ , and $\lambda _1,\ldots ,\lambda _m\geq 0$ .

We use (2.2) and (2.4) to obtain the following equivalent formulation of [Reference Hug, Mussnig and Ulivelli12, Lemma 4.6]. Here, we write $V\colon ({\mathcal K}^n)^n\to {\mathbb R}$ for the mixed volume, which is defined as the unique symmetric map such that

$$\begin{align*}V_n(\lambda_1 K_1 + \cdots + \lambda_m K_m) = \sum_{i_1,\ldots,i_n=1}^m \lambda_{i_1} \cdots \lambda_{i_n} V(K_{i_1},\ldots,K_{i_n})\end{align*}$$

for $m\in {\mathbb N}$ , $K_1,\ldots ,K_m\in {\mathcal K}^n$ , and $\lambda _1,\ldots ,\lambda _m\geq 0$ . See [Reference Schneider21, Theorem 5.1.7] for further details on mixed volumes.

Lemma 2.3 If $K_1,\ldots ,K_n\in {\mathcal K}^n$ , then

$$\begin{align*}\textrm{MA}^{\!*}(\mathbf{I}_{K_1},\ldots,\mathbf{I}_{K_n};B)= V(K_1,\ldots,K_n) \delta_o(B)\end{align*}$$

for Borel sets $B\subseteq {\mathbb R}^n$ , where $\delta _o$ is the Dirac measure at the origin. In particular,

$$\begin{align*}\binom{n}{j} \textrm{MA}^{\!*}(\mathbf{I}_K[j],\mathbf{I}_{B^n}[n-j];B)=\kappa_{n-j} V_j(K) \delta_o(B)\end{align*}$$

for $0\leq j\leq n$ .

The next result is a Kubota-type formula for conjugate mixed Monge–Ampère measures and was established by the authors in [Reference Hug, Mussnig and Ulivelli12, Theorem 5.1] (see [Reference Colesanti and Hug3] for a related result). Here, for $1\leq k\leq n$ , we denote by $\operatorname {G}(n,k)$ the Grassmannian of k-dimensional linear subspaces of ${\mathbb R}^n$ , and integration on this space is always understood with respect to the Haar probability measure. For $u\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ and $E\in \operatorname {G}(n,k)$ , we write

$$\begin{align*}\operatorname{proj}_E u(x_E) = \min\nolimits_{y\in E^\perp} u(x_E +y)\end{align*}$$

with $x_E\in E$ , for the projection function of u.

Lemma 2.4 If $1\leq k <n$ and $\varphi \colon {\mathbb R}^n\to [0,\infty )$ is measurable, then

$$ \begin{align*} & \frac{1}{\kappa_n} \int_{{\mathbb R}^n} \varphi(y) \,\mathrm{d} \textrm{MA}^{\!*}(u_1,\ldots,u_k,\mathbf{I}_{B^n}[n-k];y)\\ &\quad =\frac{1}{\kappa_k} \int_{\operatorname{G}(n,k)}\int_E \varphi(y_E) \,\mathrm{d}\textrm{MA}^{\!*}_E(\operatorname{proj}_E u_1,\ldots,\operatorname{proj}_E u_k;y_E) \,\mathrm{d} E \end{align*} $$

for $u_1,\ldots ,u_k\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ .

For $t\geq 0$ , let $u_t\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ be defined by $u_t(x)=t\vert x\vert + \mathbf {I}_{B^n}(x)$ for $x\in {\mathbb R}^n$ . We need the following result, which is a consequence of [Reference Colesanti, Ludwig and Mussnig6, Lemma 8.4] together with the defining relation (2.4).

Lemma 2.5 If $1\leq j \leq n$ and $\alpha \in C_c([0,\infty ))$ , then

$$\begin{align*}\int_{{\mathbb R}^n}\alpha(|y|) \,\mathrm{d}\textrm{MA}^{\!*}(u_t[j],\mathbf{I}_{B^n}[n-j];y) = \kappa_n \alpha(t)\end{align*}$$

for $t\geq 0$ .

Lastly, we need some results on valuations on ${\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ , which are defined analogously to (1.4). By [Reference Colesanti, Ludwig and Mussnig5, Proposition 3.5], a map $\operatorname {\mathrm {\operatorname {Z}}}\colon {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)} \to {\mathbb R}$ is a valuation if and only if $v\mapsto \operatorname {\mathrm {\operatorname {Z}}}^*(v)=\operatorname {\mathrm {\operatorname {Z}}}(v^*)$ is a valuation on ${\mbox {Conv}({\mathbb R}^n; {\mathbb R})}$ . We say that $\operatorname {\mathrm {\operatorname {Z}}}$ is epi-translation invariant if $\operatorname {\mathrm {\operatorname {Z}}}^*$ is dually epi-translation invariant or equivalently if $\operatorname {\mathrm {\operatorname {Z}}}(u\circ \tau ^{-1}+c)=\operatorname {\mathrm {\operatorname {Z}}}(u)$ for $u\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ , translations $\tau $ on ${\mathbb R}^n$ , and $c\in {\mathbb R}$ . The operator $\operatorname {\mathrm {\operatorname {Z}}}$ is epi-homogeneous of degree $j\in {\mathbb N}$ if for $u\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ and $\lambda \geq 0$ .

The following result from [Reference Colesanti, Ludwig and Mussnig6, Proposition 5.3] provides some examples of valuations on ${\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ . We denote by $C_c({\mathbb R}^n)$ the set of continuous real-valued functions with compact support on ${\mathbb R}^n$ .

Lemma 2.6 Let $\varphi \in C_c({\mathbb R}^n)$ and $0\leq j\leq n$ . If $u_1,\ldots ,u_{n-1}\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ , then

$$\begin{align*}u\mapsto \int_{{\mathbb R}^n} \varphi(x) \,\mathrm{d}\textrm{MA}^{\!*}(u_1,\ldots,u_{n-j},u[j];x)\end{align*}$$

defines a continuous, epi-translation invariant valuation on ${\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ that is epi-homogeneous of degree j.

We say that a map $\operatorname {\mathrm {\operatorname {Z}}}$ on ${\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ is rotation invariant if $\operatorname {\mathrm {\operatorname {Z}}}(u\circ \vartheta ^{-1})=\operatorname {\mathrm {\operatorname {Z}}}(u)$ for $u\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ and $\vartheta \in \operatorname {SO}(n)$ . The following Hadwiger-type result, provided in [Reference Colesanti, Ludwig and Mussnig8, Theorem 1.3], is equivalent to Theorem 1.3 and shows that not many examples of valuations remain under the additional assumption of rotation invariance. For the version stated below, see [Reference Colesanti, Ludwig and Mussnig6, Theorem 1.7].

Theorem 2.7 A functional $\operatorname {\mathrm {\operatorname {Z}}}\colon {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}\to {\mathbb R}$ is a continuous, epi-translation and rotation invariant valuation if and only if there exist functions $\alpha _0,\ldots ,\alpha _n\in C_c({[0,\infty )})$ such that

$$\begin{align*}\operatorname{\mathrm{\operatorname{Z}}}(u)=\sum_{j=0}^n \int_{{\mathbb R}^n} \alpha_j(|y|)\,\mathrm{d}\textrm{MA}^{\!*}(u[j],\mathbf{I}_{B^n}[n-j];y)\end{align*}$$

for $u\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ .

Let us remark that Lemma 2.5 shows that the operator $\operatorname {\mathrm {\operatorname {Z}}}$ that appears in Theorem 2.7 uniquely determines the densities $\alpha _j$ .

3 Proof of Theorem 1.4

Throughout this section, we use the abbreviated notation

$$\begin{align*}\textrm{MA}^{\!*}_j(u;\cdot) = \textrm{MA}^{\!*}(u[j],\mathbf{I}_{B^n}[n-j];\cdot)=\textrm{MA}(u^*[j],h_{B^n}[n-j];\cdot)\end{align*}$$

for $0\leq j\leq n$ and $u\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ , which was introduced in [Reference Colesanti, Ludwig and Mussnig6].

To prove the next result, we follow the strategy of the proof of [Reference Colesanti, Ludwig and Mussnig9, Lemma 3.4]. Moreover, we use that $(u\circ \vartheta )^*=u^*\circ \vartheta $ , for $u\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ and $\vartheta \in \operatorname {SO}(n)$ , and $\textrm {MA}(v;\vartheta B)=\textrm {MA}(v\circ \vartheta ;B)$ , for $v\in {\mbox {Conv}({\mathbb R}^n; {\mathbb R})}$ , $\vartheta \in \operatorname {SO}(n)$ , and Borel sets $B \subseteq {\mathbb R}^n$ .

Lemma 3.1 For any $1\leq j\leq n$ , fixed $\bar {v}\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ and $\alpha \in C_c([0,\infty ))$ the map

(3.1) $$ \begin{align} u\mapsto \int_{\operatorname{SO}(n)} \int_{{\mathbb R}^n} \alpha(|y|) \,\mathrm{d}\textrm{MA}^{\!*}_j\big(u\mathbin{\Box} (\bar{v}\circ\vartheta^{-1});y\big) \,\mathrm{d}\vartheta \end{align} $$

is a continuous, epi-translation and rotation invariant valuation on ${\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ .

Proof We start by showing that

(3.2) $$ \begin{align} (\vartheta,u)\mapsto \int_{{\mathbb R}^n} \alpha(|y|) \,\mathrm{d}\textrm{MA}^{\!*}_j\big(u\mathbin{\Box} (\bar{v}\circ\vartheta^{-1});y\big) \end{align} $$

is jointly continuous on $\operatorname {SO}(n)\times {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ . For this, observe that it follows from the rotational symmetry of the integrand and the two basic facts mentioned before the statement of the lemma that

$$ \begin{align*} \int_{{\mathbb R}^n} \alpha(|y|) \,\mathrm{d}\textrm{MA}^{\!*}_j\big(u\mathbin{\Box} (\bar{v}\circ\vartheta^{-1});y\big) &= \int_{{\mathbb R}^n} \alpha(|y|) \,\mathrm{d}\textrm{MA}^{\!*}_j\big((u\mathbin{\Box} (\bar{v}\circ\vartheta^{-1}))\circ \vartheta ;y\big)\\ &= \int_{{\mathbb R}^n} \alpha(|y|) \,\mathrm{d}\textrm{MA}^{\!*}_j\big((u\circ \vartheta)\mathbin{\Box} \bar{v} ;y\big) \end{align*} $$

for every $u\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ and $\vartheta \in \operatorname {SO}(n)$ . Thus, by Lemma 2.2, Lemma 2.6, and Lemma 2.1, the map given in (3.2) is jointly continuous on $\operatorname {SO}(n)\times {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ .

Let $u_i\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ , $i\in {\mathbb N}$ , be such that $u_i$ epi-converges to some $\bar {u}\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ , which means that $\{u_i : i\in {\mathbb N}\} \cup \{\bar {u}\}$ is sequentially compact. Together with the fact that $\operatorname {SO}(n)$ is compact and the map given in (3.2) is jointly continuous, it follows that the supremum

(3.3) $$ \begin{align} \sup\left\{\left\vert \int_{{\mathbb R}^n} \alpha(|y|) \,\mathrm{d}\textrm{MA}^{\!*}_j\big(u_i\mathbin{\Box} (\bar{v}\circ\vartheta^{-1});y\big) \right \vert: i\in{\mathbb N}, \vartheta \in \operatorname{SO}(n) \right\} \end{align} $$

is finite. Thus, we may apply the dominated convergence theorem to obtain

$$ \begin{align*} & \lim_{i\to\infty} \int_{\operatorname{SO}(n)} \int_{{\mathbb R}^n} \alpha(|y|) \,\mathrm{d}\textrm{MA}^{\!*}_j \big(u_i\mathbin{\Box} (\bar{v}\circ\vartheta^{-1});y\big) \,\mathrm{d}\vartheta\\ &\quad = \int_{\operatorname{SO}(n)} \int_{{\mathbb R}^n} \alpha(|y|) \,\mathrm{d}\textrm{MA}^{\!*}_j\big(\bar{u}\mathbin{\Box} (\bar{v}\circ\vartheta^{-1});y\big) \,\mathrm{d}\vartheta, \end{align*} $$

which shows that (3.1) is continuous.

Lastly, it follows from the properties of infimal convolution and the corresponding properties provided in Lemma 2.6 that (3.1) defines an epi-translation and rotation invariant valuation.

For the proof of the main result of this section, we need the elementary property

(3.4) $$ \begin{align} \textrm{MA}^{\!*}(\mathbf{I}_{B^n};\cdot) = \kappa_n \delta_{o}, \end{align} $$

which is a special case of Lemma 2.3. The next result is the equivalent version of Theorem 1.4 on ${\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ .

Theorem 3.3 If $0\leq j\leq n$ and $\alpha \colon [0,\infty )\to [0,\infty )$ is measurable, then

(3.5) $$ \begin{align} & \kappa_n \int_{\operatorname{SO}(n)} \int_{{\mathbb R}^n} \alpha(|y|) \,\mathrm{d}\textrm{MA}^{\!*}_j\big(u\mathbin{\Box}(v\circ \vartheta^{-1});y\big) \,\mathrm{d}\vartheta\notag\\ &\quad =\sum_{i=0}^j \binom{j}{i} \int_{{\mathbb R}^n}\int_{{\mathbb R}^n} \alpha(\max\{|x|,|y|\})\,\mathrm{d}\textrm{MA}^{\!*}_{j-i}(v;y)\,\mathrm{d}\textrm{MA}^{\!*}_i(u;x) \end{align} $$

for $u,v\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ .

Proof For the proof, it is sufficient to consider a function $\alpha \in C_c({[0,\infty )})$ . Indeed, for given $u,v\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ , all occurring integrals in (3.5) are finite if $\alpha $ is continuous with compact support. Thus, both sides of (3.5) define positive linear functionals on $C_c({[0,\infty )})$ . By the Riesz representation theorem (see, for example, [Reference Cohn2, Theorem 7.2.8]), there exist unique Borel measures $\mu _1,\mu _2$ on $[0,\infty )$ , (which are, in fact, Radon measures, since they are finite on compact sets) such that

$$\begin{align*}\kappa_n \int_{\operatorname{SO}(n)} \int_{{\mathbb R}^n} \alpha(|y|) \,\mathrm{d}\textrm{MA}^{\!*}_j\big(u\mathbin{\Box}(v\circ \vartheta^{-1});y\big) \,\mathrm{d}\vartheta = \int_{[0,\infty)} \alpha(z) \,\mathrm{d}\mu_1(z) \end{align*}$$

for every $\alpha \in C_c({[0,\infty )})$ , and, similarly, the right side of (3.5) can be written as $\int _{[0,\infty )} \alpha \,\mathrm {d}\mu _2$ . By (3.5) together with the uniqueness part of the Riesz representation theorem, $\mu _1$ and $\mu _2$ must be equal, and therefore equality holds in (3.5) when integrating an arbitrary nonnegative Borel measurable function.

If $j=0$ , the statement trivially follows from (3.4). Thus, we will assume $1\leq j \leq n$ throughout the following. For $u,v\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ , set

(3.6) $$ \begin{align} \operatorname{\mathrm{\operatorname{Z}}}(u,v)=\int_{\operatorname{SO}(n)} \int_{{\mathbb R}^n} \alpha(|y|) \,\mathrm{d}\textrm{MA}^{\!*}_j\big(u\mathbin{\Box}(v\circ \vartheta^{-1});y\big) \,\mathrm{d}\vartheta. \end{align} $$

For fixed $\bar {v}\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ , it follows from Lemma 3.1 that $u\mapsto \operatorname {\mathrm {\operatorname {Z}}}(u,\bar {v})$ is a continuous, epi-translation and rotation invariant valuation on ${\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ . Thus, by Theorem 2.7, there exist functions $\alpha _{i,\bar {v}}\in C_c([0,\infty ))$ , $0\leq i\leq n$ , such that

$$\begin{align*}\operatorname{\mathrm{\operatorname{Z}}}(u,\bar{v})= \sum_{i=0}^n \int_{{\mathbb R}^n} \alpha_{i,\bar{v}}(|y|)\,\mathrm{d}\textrm{MA}^{\!*}_i(u;y)\end{align*}$$

for every $u\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ . In particular, if we choose , where $u_t=t|\cdot |+\mathbf {I}_{B^n}$ , it follows from (3.4), Lemma 2.6, and Lemma 2.5 that

(3.7)

for every $t,\lambda \geq 0$ . Since for every fixed $\bar {u}\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ also the map $v\mapsto \operatorname {\mathrm {\operatorname {Z}}}(\bar {u},v)$ is a continuous, epi-translation and rotation invariant valuation, it follows from (3.7) together with homogeneity that for every fixed $\bar {t}\geq 0$ , each of the maps

$$\begin{align*}v\mapsto \alpha_{0,{v}}(0)\quad\text{and}\quad v\mapsto \alpha_{i,v}(\bar{t}),\;1\leq i\leq n\end{align*}$$

is a continuous, epi-translation and rotation invariant valuation. Thus, by Theorem 2.7, there exist functions $\alpha _{0,k}\in C_c([0,\infty )), \alpha _{i,k}(\bar {t},\cdot )\in C_c([0,\infty ))$ , $1\leq i\leq n$ , $0\leq k\leq n$ , such that

$$\begin{align*}\alpha_{0,v}(0)=\sum_{k=0}^n \int_{{\mathbb R}^n} \alpha_{0,k}(|y|) \,\mathrm{d}\textrm{MA}^{\!*}_k(v;y)\end{align*}$$

and

$$\begin{align*}\alpha_{i,v}(\bar{t}) =\sum_{k=0}^n \int_{{\mathbb R}^n} \alpha_{i,k}(\bar{t},|y|)\,\mathrm{d}\textrm{MA}^{\!*}_k(v;y)\end{align*}$$

for every $v\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ and $1\leq i\leq n$ . We thus have

(3.8) $$ \begin{align} \begin{aligned} \operatorname{\mathrm{\operatorname{Z}}}(u,v)&= \kappa_n \alpha_{0,v}(0)+ \sum_{i=1}^n \int_{{\mathbb R}^n} \alpha_{i,v}(|x|)\,\mathrm{d}\textrm{MA}^{\!*}_i(u;x)\\ &= \kappa_n \sum_{k=0}^n \int_{{\mathbb R}^n}\alpha_{0,k}(|y|)\,\mathrm{d}\textrm{MA}^{\!*}_k(v;y)\\ &\quad +\sum_{i=1}^n\sum_{k=0}^n \int_{{\mathbb R}^n} \int_{{\mathbb R}^n} \alpha_{i,k}(|x|,|y|) \,\mathrm{d}\textrm{MA}^{\!*}_k(v;y)\,\mathrm{d}\textrm{MA}^{\!*}_i(u;x) \end{aligned} \end{align} $$

for every $u,v\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ .

It remains to determine the relation between $\alpha _{i,k}$ and $\alpha $ . In order to do this, we will evaluate $\operatorname {\mathrm {\operatorname {Z}}}(u,v)$ at and with $\lambda ,\mu ,s,t\geq 0$ . Notice that

for $\vartheta \in \operatorname {SO}(n)$ and $x\in {\mathbb R}^n$ . For $0\le s\le t$ , we now have

Hence,

for every $\vartheta \in \operatorname {SO}(n)$ and $E\in \operatorname {G}(n,j)$ . Thus, it follows from (3.6) and Lemma 2.4 (for $j=n$ the lemma holds trivially) that

(3.9)

for $0\leq s\leq t$ . However, by (3.8), (3.4), and Lemma 2.5,

(3.10)

Since $\lambda ,\mu \geq 0$ were arbitrary, we can compare coefficients of the last two equations to obtain

$$\begin{align*}\kappa_n \alpha_{j,0}(s,0)=\alpha(s),\quad \kappa_n \alpha_{0,j}(t)=\alpha(t),\quad \kappa_n \alpha_{i,j-i}(s,t)=\binom{j}{i}\alpha(t), \end{align*}$$

for $1\leq i \leq j-1$ and $\alpha _{i,k}(s,t)=0$ if $i+k\neq j$ whenever $0\leq s \leq t$ .

Proceeding for the case $s\ge t$ similarly as in the derivation of (3.9), we obtain

A comparison with the coefficients available from (3.10) shows that

$$\begin{align*}\kappa_n \alpha_{j,0}(s,0)=\alpha(s), \quad \kappa_n \alpha_{0,j}(t)=\alpha(t),\quad \kappa_n \alpha_{i,j-i}(s,t)=\binom{j}{i}\alpha(s),\end{align*}$$

for $1\leq i \leq j-1$ and $\alpha _{i,k}(s,t)=0$ if $i+k\neq j$ , also for $s\ge t$ .

The claim now follows after considering (3.4).

4 Further formulas

For $\alpha \in C_c({[0,\infty )})$ and $0\leq j\leq n$ , we define the jth functional intrinsic volume on ${\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ with density $\alpha $ , denoted by $\overline {\operatorname {V}}_{j,\alpha }$ , as

(4.1) $$ \begin{align} \overline{\operatorname{V}}_{j,\alpha}(u)=\binom{n}{j}\frac{1}{\kappa_{n-j}} \int_{{\mathbb R}^n} \alpha(|y|)\,\mathrm{d}\textrm{MA}^{\!*}_j(u;y) \end{align} $$

for $u\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ . Clearly, $\overline {\operatorname {V}}_{j,\alpha }^*(v)=\overline {\operatorname {V}}_{j,\alpha }(v^*)$ for every $v\in {\mbox {Conv}({\mathbb R}^n; {\mathbb R})}$ .

By (4.1), we have the following reformulation of Theorem 3.3.

Theorem 4.1 If $0\leq j\leq n$ and $\alpha \in C_c([0,\infty ))$ , then

(4.2) $$ \begin{align} \kappa_n \kappa_{n-j} &\int_{\operatorname{SO}(n)} \overline{\operatorname{V}}_{j,\alpha}\big(u\mathbin{\Box} (v\circ\vartheta^{-1})\big) \,\mathrm{d}\vartheta\notag\\ &\qquad =\binom{n}{j} \sum_{i=0}^j \binom{j}{i} \int_{{\mathbb R}^n}\int_{{\mathbb R}^n} \alpha(\max\{|x|,|y|\})\,\mathrm{d}\textrm{MA}^{\!*}_{j-i}(v;y)\,\mathrm{d}\textrm{MA}^{\!*}_i(u;x) \end{align} $$

for $u,v\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ .

If in (4.2) we choose v to be the convex indicator function $\mathbf {I}_L$ of a convex body $L \in {\mathcal K}^n$ , then a direct application of Lemma 2.3 gives a specialization of Theorem 4.1 reading as follows, where we write

$$\begin{align*}\genfrac{[}{]}{0pt}{}{k}{j}=\binom{k}{j}\frac{\kappa_k}{\kappa_j \kappa_{k-j}} \end{align*}$$

for $j,k\in {\mathbb N}$ .

Corollary 4.2 If $0\leq j\leq n$ and $\alpha \in C_c([0,\infty ))$ , then

$$ \begin{align*} \int_{\operatorname{SO}(n)}\overline{\operatorname{V}}_{j,\alpha} (u \mathbin{\Box} \mathbf{I}_{\vartheta L}) \,\mathrm{d} \vartheta =\sum_{i=0}^j \genfrac{[}{]}{0pt}{}{n-i}{j-i}{\genfrac{[}{]}{0pt}{}{n}{j-i}}^{-1}\overline{\operatorname{V}}_{i,\alpha} (u) V_{j-i}(L) \end{align*} $$

for $u \in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ and $L \in {\mathcal K}^n$ .

Proof It follows from Lemma 2.3 that for $u \in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ , $L \in {\mathcal K}^n$ , and $0\leq j \leq n$ , we have

$$\begin{align*}\binom{n}{j}\textrm{MA}^{\!*}_j(\mathbf{I}_L;B)=\kappa_{n-j}V_j(L)\delta_0(B)\end{align*}$$

for Borel sets $B \subseteq {\mathbb R}^n$ . Applying Theorem 4.1, we then infer

$$ \begin{align*} \int_{\operatorname{SO}(n)}&\overline{\operatorname{V}}_{j,\alpha}(u\mathbin{\Box} I_{\vartheta L})\,\mathrm{d} \vartheta\\ &=\frac{\binom{n}{j}}{\kappa_n \kappa_{n-j}} \sum_{i=0}^j \binom{j}{i} \int_{{\mathbb R}^n} \int_{{\mathbb R}^n} \alpha(\max \{|x|,|y|\})\,\mathrm{d} \textrm{MA}^{\!*}_{j-i}(\mathbf{I}_L;y)\,\mathrm{d} \textrm{MA}^{\!*}_i(u;x)\\ &=\frac{\binom{n}{j}}{\kappa_n \kappa_{n-j}}\sum_{i=0}^j \frac{\binom{j}{i}\kappa_{n-(j-i)}}{\binom{n}{j-i}} V_{j-i}(L)\int_{{\mathbb R}^n}\alpha(|x|)\,\mathrm{d} \textrm{MA}^{\!*}_i(u;x)\\ &=\sum_{i=0}^j\frac{\binom{n}{j}\binom{j}{i}}{\binom{n}{j-i}\binom{n}{i}}\frac{\kappa_{n-i}\kappa_{n-(j-i)}}{\kappa_n \kappa_{n-j}}\overline{\operatorname{V}}_{i,\alpha}(u)V_{j-i}(L)\\ &=\sum_{i=0}^j \genfrac{[}{]}{0pt}{}{n-i}{j-i}{\genfrac{[}{]}{0pt}{}{n}{j-i}}^{-1}\overline{\operatorname{V}}_{i,\alpha} (u) V_{j-i}(L), \end{align*} $$

where we used that $\alpha (\max \{|x|,0\})=\alpha (|x|)$ for $x\in {\mathbb R}^n$ and

$$\begin{align*}\frac{\binom{n}{j}\binom{j}{i}}{\binom{n}{i}}=\binom{n-i}{j-i} \end{align*}$$

for $0\leq i\leq j\leq n$ .

Note that if in the last result we furthermore also choose u to be the indicator function of a convex body $K\in {\mathcal K}^n$ , then we recover the additive kinematic formula for convex bodies (1.1), which can be written as

$$\begin{align*}\int_{\operatorname{SO}(n)} V_j(K+\vartheta L)\,\mathrm{d}\vartheta=\sum_{i=0}^j \genfrac{[}{]}{0pt}{}{n-i}{j-i}{\genfrac{[}{]}{0pt}{}{n}{j-i}}^{-1}V_i(K) V_{j-i}(L). \end{align*}$$

Next, we consider mixed functionals. A corollary of (1.1) for the mixed volume $V:({\mathcal K}^n)^n\to {\mathbb R}$ states that if $0\leq j\leq n$ , then

(4.3) $$ \begin{align} \binom{n}{j} \int_{\operatorname{SO}(n)} V(K[j],\vartheta L [n-j]) \,\mathrm{d} \vartheta = \genfrac{[}{]}{0pt}{}{n}{j}^{-1} V_j(K) V_{n-j}(L) \end{align} $$

for every $K,L\in {\mathcal K}^n$ . See, for example, formula (6.7) in [Reference Schneider and Weil22].

For $\alpha \in C_c({[0,\infty )})$ , we define the operator $\operatorname {\mathrm {\overline {\operatorname {V}}}}_\alpha $ on $({\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)})^n$ as

$$\begin{align*}\operatorname{\mathrm{\overline{\operatorname{V}}}}_\alpha(u_1,\ldots,u_n)= \int_{{\mathbb R}^n} \alpha(|y|)\,\mathrm{d} \textrm{MA}^{\!*}(u_1,\ldots,u_n;y) \end{align*}$$

for $u_1,\ldots ,u_n\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ . Clearly, by the properties of the conjugate mixed Monge–Ampère measure, the functional $\operatorname {\mathrm {\overline {\operatorname {V}}}}_\alpha $ is symmetric in its entries. Moreover, in each of its arguments, it is continuous with respect to epi-convergence, epi-homogeneous of degree 1, and epi-translation invariant. We remark that functionals of this form were also treated in [Reference Alesker1] and, from a valuation point of view, in [Reference Colesanti, Ludwig and Mussnig5, Reference Knoerr16].

By Lemma 2.3, it is immediate to check that $\operatorname {\mathrm {\overline {\operatorname {V}}}}_\alpha $ generalizes the classical mixed volumes; that is,

$$\begin{align*}\operatorname{\mathrm{\overline{\operatorname{V}}}}_\alpha(\mathbf{I}_{K_1},\ldots,\mathbf{I}_{K_n})=\alpha(0)V(K_1,\dots,K_n) \end{align*}$$

for $K_1,\dots ,K_n \in {\mathcal K}^n$ . We have the following functional integral formula which includes (4.3) in the special case where $k=n$ , $u=\mathbf {I}_K$ , and $v=\mathbf {I}_L$ (see also [Reference Hug and Weil14, Lemma 5.8]).

Corollary 4.3 If $0\leq j\leq k\le n$ and $\alpha \in C_c({[0,\infty )})$ , then

$$ \begin{align*} &\int_{\operatorname{SO}(n)} \operatorname{\mathrm{\overline{\operatorname{V}}}}_{\alpha}\big(u_1,\ldots,u_j,v_1\circ \vartheta^{-1},\ldots,v_{k-j}\circ \vartheta^{-1},\mathbf{I}_{B^n}[n-k]\big)\,\mathrm{d}\vartheta\\ &\;= \frac{1}{\kappa_n} \int_{{\mathbb R}^n}\int_{{\mathbb R}^n} \alpha(\max\{|x|,|y|\}) \,\mathrm{d}\textrm{MA}^{\!*}_{j}(u_1,\ldots,u_j;x) \,\mathrm{d} \textrm{MA}^{\!*}_{k-j}(v_1,\ldots,v_{k-j};y) \end{align*} $$

for $u_1,\ldots ,u_j,v_1,\ldots ,v_{k-j}\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ .

Proof Let $u,v\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ . For $\varepsilon>0$ , it follows from Theorem 4.1 and the properties of the measures $\textrm {MA}^{\!*}_{n-j}(v;\cdot )$ that

for every $u,v\in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ . However, by (2.5), we have

The result now follows after comparing coefficients together with multilinearity.

As an application, Corollary 4.3 can be used to obtain generalizations of further formulas. In particular, mimicking the so-called Minkowski difference (see, for example, [Reference Schneider and Weil22, Note 3 of Section 6.1]), we can introduce the operation of inf-deconvolution. If for $u,v \in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ , there exists $w \in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ such that

$$\begin{align*}w \mathbin{\Box} v = u, \end{align*}$$

then we say that w is the inf-deconvolution of u and v, which we denote by

$$\begin{align*}w=u \mathbin{\diamond} v. \end{align*}$$

Equivalently, this means that $u\mathbin {\diamond } v$ exists if and only if the (pointwise) difference $u^*-v^*$ is an element of ${\mbox {Conv}({\mathbb R}^n; {\mathbb R})}$ and

$$\begin{align*}(u\mathbin{\diamond} v)^*=u^*-v^*.\end{align*}$$

Moreover, we say that v rolls freely in u if for every $\vartheta \in \operatorname {SO}(n)$ , the expression $u \mathbin {\diamond } (v \circ \vartheta ^{-1})$ is well-defined. With this new terminology at hand, we obtain the following consequence of Corollary 4.3.

Corollary 4.4 Let $0\le k \le n$ , $\alpha \in C_c({[0,\infty )})$ , and $u,v \in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^n)}$ . If v rolls freely in u, then

(4.4) $$ \begin{align} \begin{aligned} & \int_{\operatorname{SO}(n)} \overline{\operatorname{V}}_{k,\alpha}\big(u \mathbin{\diamond} (v\circ \vartheta^{-1})\big) d\vartheta\\ &\quad =\frac{\binom{n}{k}}{\kappa_n\kappa_{n-k}} \sum_{j=0}^k(-1)^{k-j} \binom{k}{j} \int_{{\mathbb R}^n} \int_{{\mathbb R}^n} \alpha(\max\{|x|,|y|\})\,\mathrm{d} \textrm{MA}^{\!*}_{k-j}(v;y)\,\mathrm{d}\textrm{MA}^{\!*}_{j}(u;x). \end{aligned} \end{align} $$

Proof Since $\operatorname {\mathrm {\overline {\operatorname {V}}}}_\alpha $ is linear with respect to inf-convolution in each of its arguments. and presuming that $u\mathbin {\diamond } (v\circ \vartheta ^{-1})$ exists for every $\vartheta \in \operatorname {SO}(n)$ , we have

$$ \begin{align*} &\overline{\operatorname{V}}_{k,\alpha}\big(u \mathbin{\diamond} (v\circ \vartheta^{-1})\big)\\ &=\operatorname{\mathrm{\overline{\operatorname{V}}}}_\alpha\big(u \mathbin{\diamond} (v\circ \vartheta^{-1})[k],\mathbf{I}_{B^{n}}[n-k]\big)\\ &=\operatorname{\mathrm{\overline{\operatorname{V}}}}_\alpha\big(u \mathbin{\diamond} (v\circ \vartheta^{-1})[k-1],u,\mathbf{I}_{B^{n}}[n-k]\big)\\ &\qquad -\operatorname{\mathrm{\overline{\operatorname{V}}}}_{\alpha}\big(u \mathbin{\diamond} (v\circ \vartheta^{-1})[n-1],(v\circ \vartheta^{-1}),\mathbf{I}_{B^{n}}[n-k]\big)\\ &\;\;\vdots\\ &=\sum_{j=0}^k (-1)^{k-j}\binom{k}{j}\operatorname{\mathrm{\overline{\operatorname{V}}}}_\alpha\big(u[j],(v\circ \vartheta^{-1})[k-j],\mathbf{I}_{B^{n}}[n-k]\big), \end{align*} $$

which can be proved in detail by induction on $k\in \{0,\ldots ,n\}$ . Integration over $\operatorname {SO}(n)$ together with an application of Corollary 4.3 results in relation (4.4).

5 Singular Hessian integrals

In this section, we demonstrate another application of the special case $k=n$ of Corollary 4.3. Let us first state its equivalent version on ${\mbox {Conv}({\mathbb R}^n; {\mathbb R})}$ (for ${k=n}$ ), where we write $\textrm {MA}_j(v;\cdot )=\textrm {MA}(v[j],h_{B^n}[n-j];\cdot )$ for $v\in {\mbox {Conv}({\mathbb R}^n; {\mathbb R})}$ and $0\leq j\leq n$ .

Corollary 5.1 If $0\leq j\leq n$ and $\alpha \in C_c({[0,\infty )})$ , then

$$ \begin{align*} & \int_{\operatorname{SO}(n)} \int_{{\mathbb R}^n} \alpha(|x|)\,\mathrm{d}\textrm{MA} \big(v[j],w\circ\vartheta^{-1}[n-j];x\big) \,\mathrm{d}\vartheta\\ &\quad=\frac{1}{\kappa_n} \int_{{\mathbb R}^n}\int_{{\mathbb R}^n} \alpha(\max\{|x|,|y|\})\,\mathrm{d}\textrm{MA}_{n-j}(w;y)\,\mathrm{d}\textrm{MA}_j(v;x) \end{align*} $$

for $v,w\in {\mbox {Conv}({\mathbb R}^n; {\mathbb R})}$ .

As mentioned in Section 1, functional intrinsic volumes were previously defined in terms of Hessian measures. For this, let

$$\begin{align*}D_j^n=\left\{\zeta\in C_b((0,\infty)) : \lim_{s\to 0^+} s^{n-j}\zeta(s)=0, \exists \lim_{s\to 0^+} \int_s^\infty t^{n-j-1}\zeta(t)\,\mathrm{d} t \in{\mathbb R} \right\}\end{align*}$$

for $0\leq j\leq n-1$ , where $C_b((0,\infty ))$ denotes the set of continuous function with bounded support on $(0,\infty )$ . In addition, set

$$\begin{align*}D_n^n=\left\{\zeta\in C_b((0,\infty)): \exists \lim_{s\to 0^+} \zeta(s) \in {\mathbb R}\right\},\end{align*}$$

which we identify with $C_c({[0,\infty )})$ . In [Reference Colesanti, Ludwig and Mussnig8, Theorem 1.4] and later also in [Reference Colesanti, Ludwig and Mussnig9, Reference Colesanti, Ludwig and Mussnig6, Reference Knoerr17], it was shown that for $0\leq j\leq n$ and $\zeta \in D_j^n$ , the map

(5.1) $$ \begin{align} v\mapsto \int_{{\mathbb R}^n} \zeta(|x|) [{\operatorname{D}}^2 v(x)]_j \,\mathrm{d} x \end{align} $$

continuously extends from ${\mbox {Conv}({\mathbb R}^n; {\mathbb R})}\cap C_+^2({\mathbb R}^n)$ to ${\mbox {Conv}({\mathbb R}^n; {\mathbb R})}$ . This extension was used as the original definition for functional intrinsic volumes, meaning they can be understood as singular Hessian integrals.

In [Reference Colesanti, Ludwig and Mussnig6], an alternative proof of existence for the continuous extension of (5.1) was found. The essential observation (see [Reference Colesanti, Ludwig and Mussnig6, Proposition 6.7]) is that

(5.2) $$ \begin{align} \int_{{\mathbb R}^n} \zeta(|x|)\,\mathrm{d}\textrm{MA}(v[j],q[n-j];x) = \int_{{\mathbb R}^n} \operatorname{\mathcal{R}}^{n-j} \zeta(|x|)\,\mathrm{d}\textrm{MA}_j(v;x) \end{align} $$

for $0\leq j\leq n-1$ , $\zeta \in D_j^n$ , and $v\in C^2({\mathbb R}^n)$ , where

$$\begin{align*}\operatorname{\mathcal{R}}^{n-j} \zeta(s) = s^{n-j} \zeta(s) + (n-j) \int_s^\infty t^{n-j-1}\zeta(t) \,\mathrm{d} t \end{align*}$$

for $s>0$ and where $q(x)=|x|^2/2$ . If we consistently define $\operatorname {\mathcal {R}}^0\zeta =\zeta $ for $\zeta \in D^n_n$ , then (5.2) remains true also for $j=n$ . In addition, it was previously shown in [Reference Colesanti, Ludwig and Mussnig9, Lemma 3.8], that $\operatorname {\mathcal {R}}^{n-j}$ is a bijection from $D_j^n$ to $D_n^n$ . Together with

(5.3) $$ \begin{align} \binom{n}{j}\,\mathrm{d}\textrm{MA}(v[j],q[n-j];x)=[{\operatorname{D}}^2 v(x)]_j \,\mathrm{d} x \end{align} $$

for $v\in {\mbox {Conv}({\mathbb R}^n; {\mathbb R})}\cap C^2({\mathbb R}^n)$ and $0\leq j\leq n$ , this then implies that (5.1) continuously extends to ${\mbox {Conv}({\mathbb R}^n; {\mathbb R})}$ (for $j=0$ , we use the convention $[{\operatorname {D}}^2 v(x)]_0\equiv 1$ ).

As we illustrate in the following, Corollary 5.1 gives a straightforward understanding of (5.2). Indeed, if in Corollary 5.1 we choose $w=q$ , then it follows from the rotational symmetry of q that

(5.4) $$ \begin{align} \int_{{\mathbb R}^n} \zeta(|x|) \,\mathrm{d}\textrm{MA}(v[j],q[n-j];x) = \int_{{\mathbb R}^n} \beta(|x|) \,\mathrm{d}\textrm{MA}_j(v;x) \end{align} $$

for every $0\leq j\leq n$ , $\zeta \in D_n^n$ , and $v\in {\mbox {Conv}({\mathbb R}^n; {\mathbb R})}$ , where

$$\begin{align*}\beta(|x|)=\frac{1}{\kappa_n} \int_{{\mathbb R}^n} \zeta(\max\{|x|,|y|\}) \,\mathrm{d}\textrm{MA}_{n-j}(q;y)\end{align*}$$

for $x\in {\mathbb R}^n$ . If $j=n$ , then $\textrm {MA}_0(q;\cdot )=\kappa _n\delta _0$ (see [Reference Hug, Mussnig and Ulivelli12, Lemma 4.6]) implies that $\beta (|x|)=\zeta (|x|)$ . If $0\le j\le n-1$ , then direct calculations (see also [Reference Colesanti, Ludwig and Mussnig6, Theorem 4.5 (d)]) show that

$$\begin{align*}\,\mathrm{d}\textrm{MA}_{n-j}(q;y)=\frac{n-j}{n}\frac{1}{|y|^j} \,\mathrm{d} y\end{align*}$$

on ${\mathbb R}^n\backslash \{o\}$ and, by [Reference Hug, Mussnig and Ulivelli12, Lemma 4.3], $\textrm {MA}_{n-j}(q;\{o\})=V(\{o\}[n-j],B^n[j])=0$ . Therefore,

$$ \begin{align*} \beta(|x|)&=\frac{1}{\kappa_n} \left(\zeta(|x|) \int_{|x|B^n} \,\mathrm{d}\textrm{MA}_{n-j}(q;y) + \int_{{\mathbb R}^n\backslash |x|B^n} \zeta(|y|)\,\mathrm{d}\textrm{MA}_{n-j}(q;y) \right)\\ &= \frac{1}{\kappa_n} \left( \zeta(|x|) \frac{n-j}{n} (n \kappa_n) \int_0^{|x|} r^{n-1-j} \,\mathrm{d} r + \frac{n-j}{n} (n \kappa_n) \int_{|x|}^{\infty} \zeta(r) r^{n-1-j} \,\mathrm{d} r\right)\\ &= \zeta(|x|)|x|^{n-j} + (n-j) \int_{|x|}^{\infty} \zeta(r) r^{n-j-1}\,\mathrm{d} r\\ &= \operatorname{\mathcal{R}}^{n-j}\zeta(|x|) \end{align*} $$

for $x\in {\mathbb R}^n$ . By using the properties of the transform $\operatorname {\mathcal {R}}^{n-j}$ and of the measures ${\textrm {MA}(v[j],q[n-j];\cdot )}$ for $v\in C^2({\mathbb R}^n)$ (see, for example, [Reference Colesanti, Ludwig and Mussnig8, Lemma 3.1]), one can now extend (5.4) from $\zeta \in D_n^n$ to $\zeta \in D_j^n$ .

We remark that using the same method as above, similar relations can be obtained between integrals with respect to $\textrm {MA}_j(v;\cdot )$ and integrals with respect to $\textrm {MA}(v[j],w[n-j];\cdot )$ , where $w \in {\mbox {Conv}({\mathbb R}^n; {\mathbb R})}$ is rotationally symmetric.

6 Formulas for convex bodies

For the proof of Theorem 1.5, we use a connection between conjugate mixed Monge–Ampère measures and mixed area measures which was established in [Reference Knoerr and Ulivelli18, Section 3] and further expanded upon in [Reference Hug, Mussnig and Ulivelli12, Section 4.2].

Let $n\geq 2$ and let $\operatorname {proj}_H\colon {\mathbb R}^n\to H$ denote the orthogonal projection onto the hyperplane $H=e_n^{\perp }$ , which we will identify with ${\mathbb R}^{n-1}$ . To each convex body $K\in {\mathcal K}^n$ , we assign the function

$$\begin{align*}\lfloor K \rfloor(x) = \begin{cases} \min\{t\in{\mathbb R} : (x,t)\in K\}\quad&\text{if } x\in \operatorname{proj}_H K,\\ \infty\quad&\text{else.} \end{cases} \end{align*}$$

This defines a lower semicontinuous, convex function on ${\mathbb R}^{n-1}$ with compact domain, and in particular, $\lfloor K \rfloor $ is an element of ${\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^{n-1})}$ . Furthermore, observe that

(6.1) $$ \begin{align} \lfloor K + L \rfloor = \lfloor K \rfloor \mathbin{\Box} \lfloor L \rfloor \end{align} $$

for $K,L\in {\mathcal K}^n$ .

In the following, we denote by $\mathbb {S}^{n-1}_{-}=\{z\in {\mathbb {S}^{n-1}} \colon \langle z,e_{n}\rangle <0\}$ the lower half-sphere in ${\mathbb R}^n$ . The gnomonic projection $\operatorname {gno}\colon \mathbb {S}^{n-1}_-\to {\mathbb R}^{n-1}$ is defined by

$$\begin{align*}\operatorname{gno}(z)=\frac{(z_1,\ldots,z_{n-1})}{|z_{n}|}\end{align*}$$

for $z=(z_1,\ldots ,z_n)\in \mathbb {S}^{n-1}_-$ , with inverse

for $x\in {\mathbb R}^{n-1}$ . By [Reference Knoerr and Ulivelli18, Section 3.2] and [Reference Hug, Mussnig and Ulivelli12, Remark 4.7], we have the following reformulation of [Reference Hug, Mussnig and Ulivelli12, Corollary 4.9].

Lemma 6.1 If $\varphi \colon \mathbb {S}^{n-1}_{-} \to [0,\infty )$ is measurable, then

$$\begin{align*} & \int_{{\mathbb R}^{n-1}} \varphi\left(\operatorname{gno}^{-1}(y)\right) \,\mathrm{d}\textrm{MA}^{\!*}(\lfloor K_1\rfloor,\ldots,\lfloor K_{n-1}\rfloor ;y) \\ &\quad =\int_{\mathbb{S}_-^n} |\langle z,e_{n}\rangle| \varphi(z) \,\mathrm{d} S(K_1,\ldots,K_{n-1},z)\end{align*}$$

for $K_1,\ldots ,K_{n-1}\in {\mathcal K}^n$ .

Proof of Theorem 1.5

Let $\beta \colon [0,1]\to [0,\infty )$ be measurable and let $\tilde \beta \colon [0,1]\to [0,\infty )$ be given by $\tilde \beta (t)=t\beta (t)$ for $t\in [0,1]$ . Since $\tilde \beta (|z_n|)=0$ if $z_n=\langle z,e_n\rangle =0$ , we thus obtain

(6.2) $$ \begin{align} &\int_{{\mathbb{S}^{n-1}}} |z_n| \beta(|z_n|) \,\mathrm{d} S((K+\vartheta L) [j],B_H^{n-1}[n-1-j],z) \notag \\ &\quad = \int_{\mathbb{S}^{n-1}_-} \tilde\beta(|z_n|) \,\mathrm{d} S((K+\vartheta L) [j],B_H^{n-1}[n-1-j],z) \end{align} $$

(6.3) $$ \begin{align} & \quad\quad+ \int_{\mathbb{S}^{n-1}_+} \tilde\beta(|z_n|) \,\mathrm{d} S((K+\vartheta L)[j],B_H^{n-1}[n-1-j],z), \end{align} $$

for $\vartheta \in \operatorname {SO}(n-1)$ , where $\mathbb {S}^{n-1}_{+}=\{z\in \mathbb {S}^{n-1} \colon \langle z,e_{n}\rangle>0\}$ .

Next, we want to obtain suitable representations for the integrals in (6.2) and (6.3) so that we can apply Theorem 3.3. Notice that the integral in (6.3) can be rewritten as an integral on $\mathbb {S}^{n-1}_-$ . Indeed, if we denote by $\bar {K}$ and $\bar {L}$ the reflections of K and L through H, respectively, then (6.3) can be written as

$$\begin{align*}\int_{\mathbb{S}^{n-1}_-} \tilde\beta(|z_n|) \,\mathrm{d} S((\bar{K}+\vartheta \bar{L}) [j],B_H^{n-1}[n-1-j],z). \end{align*}$$

Here, we used that the considered reflection fixes $B^{n-1}_H$ and elements of $\operatorname {SO}(n-1)$ .

Let $u,v \in {\mbox {Conv}_{\mathrm {sc}}({\mathbb R}^{n-1})}$ be given by $u=\lfloor K \rfloor $ and $v=\lfloor L \rfloor $ . Furthermore, let the measurable function $\alpha \colon [0,\infty )\to [0,\infty )$ be defined by the relation

for $t\in [0,\infty )$ and $s\in (0,1]$ . By (6.1) and Lemma 6.1, applied with $\varphi (z)=\beta (|\langle z,e_n\rangle |)$ for $z\in \mathbb {S}^{n-1}_{-}$ , we now have

$$ \begin{align*} &\int_{\mathbb{S}^{n-1}_-} \tilde\beta(|z_n|) \,\mathrm{d} S((K+\vartheta L) [j],B_H^{n-1}[n-1-j],z)\\ &\quad=\int_{{\mathbb R}^{n-1}} \alpha(|y|)\,\mathrm{d}\textrm{MA}^{\!*}_j(u \mathbin{\Box} (v \circ \vartheta^{-1});y), \end{align*} $$

where we have used that $v \circ \vartheta ^{-1}=\lfloor \vartheta L \rfloor $ and $\lfloor B_H^{n-1}\rfloor = \mathbf {I}_{B^{n-1}}$ . For $\bar {u}=\lfloor \bar {K} \rfloor , \bar {v}=\lfloor \bar {L} \rfloor $ , we obtain analogously

$$ \begin{align*} &\int_{\mathbb{S}^{n-1}_+} \tilde\beta(|z_n|) \,\mathrm{d} S((K+\vartheta L) [j],B_H^{n-1}[n-1-j],z)\\ &\quad=\int_{\mathbb{S}^{n-1}_-} \tilde\beta (|z_n|) \,\mathrm{d} S((\bar{K}+\vartheta \bar{L}) [j],B_H^{n-1}[n-1-j],z)\\ &\quad=\int_{{\mathbb R}^{n-1}} \alpha(|y|)\,\mathrm{d}\textrm{MA}^{\!*}_j(\bar{u} \mathbin{\Box} (\bar{v} \circ \vartheta^{-1});y). \end{align*} $$

Hence, we get

(6.4) $$ \begin{align} &\int_{{\mathbb{S}^{n-1}}} \tilde\beta(|z_n|) \,\mathrm{d} S((K+\vartheta L) [j],B_H^{n-1}[n-1-j],z)\\ &\quad=\int_{{\mathbb R}^{n-1}} \alpha(|y|)\,\mathrm{d}\textrm{MA}^{\!*}_j(u \mathbin{\Box} (v \circ \vartheta^{-1});y) + \int_{{\mathbb R}^{n-1}} \alpha(|y|)\,\mathrm{d}\textrm{MA}^{\!*}_j(\bar{u} \mathbin{\Box} (\bar{v} \circ \vartheta^{-1});y).\notag \end{align} $$

We now integrate (6.4) over $\operatorname {SO}(n-1)$ with respect to the Haar probability measure. Together with Theorem 3.3, applied with respect to the ambient space ${\mathbb R}^{n-1}$ , we infer

(6.5) $$ \begin{align} & \int_{\operatorname{SO}(n-1)} \int_{\mathbb{S}^{n-1}} \tilde\beta(|z_n|) \,\mathrm{d} S((K+\vartheta L) [j],B_H^{n-1}[n-1-j],z) \,\mathrm{d}\vartheta\notag \\ &\quad= \int_{\operatorname{SO}(n-1)}\int_{{\mathbb R}^{n-1}} \alpha(|y|)\,\mathrm{d}\textrm{MA}^{\!*}_j(u \mathbin{\Box} (v \circ \vartheta^{-1});y) \,\mathrm{d}\vartheta\notag\\ &\quad\quad +\int_{\operatorname{SO}(n-1)}\int_{{\mathbb R}^{n-1}} \alpha(|y|)\,\mathrm{d}\textrm{MA}^{\!*}_j(\bar{u} \mathbin{\Box} (\bar{v} \circ \vartheta^{-1});y) \,\mathrm{d}\vartheta\notag\\ &\quad= \frac{1}{\kappa_{n-1}} \sum_{i=0}^j \binom{j}{i} \int_{{\mathbb R}^{n-1}}\int_{{\mathbb R}^{n-1}} \alpha(\max\{|x|,|y|\})\,\mathrm{d}\textrm{MA}^{\!*}_{j-i}(v;y)\,\mathrm{d}\textrm{MA}^{\!*}_i(u;x)\notag\\ &\quad\quad+ \frac{1}{\kappa_{n-1}} \sum_{i=0}^j \binom{j}{i} \int_{{\mathbb R}^{n-1}}\int_{{\mathbb R}^{n-1}} \alpha(\max\{|x|,|y|\})\,\mathrm{d}\textrm{MA}^{\!*}_{j-i}(\bar{v};y)\,\mathrm{d}\textrm{MA}^{\!*}_i(\bar{u};x)\\ &\quad= \frac{1}{\kappa_{n-1}} \sum_{i=0}^j \binom{j}{i} \int_{\mathbb{S}^{n-1}_-}\int_{\mathbb{S}^{n-1}_-} |w_n| |z_n| \beta(\min\{|w_n|,|z_n|\})\notag\\ &\quad\qquad\quad \,\mathrm{d} S(L [j-i],B_H^{n-1}[n-1-j+i],w) \,\mathrm{d} S(K [i],B_H^{n-1}[n-1-i],z)\notag\\ &\quad\quad +\frac{1}{\kappa_{n-1}} \sum_{i=0}^j \binom{j}{i} \int_{\mathbb{S}^{n-1}_-}\int_{\mathbb{S}^{n-1}_-} |w_n| |z_n| \beta(\min\{|w_n|,|z_n|\})\notag\\ &\quad\qquad\quad \,\mathrm{d} S(\bar{L} [j-i],B_H^{n-1}[n-1-j+i],w) \,\mathrm{d} S(\bar{K} [i],B_H^{n-1}[n-1-i],z). \notag \end{align} $$

In the last step, we used Lemma 6.1 together with the fact that is decreasing, and thus,

for $a,b\in (0,1]$ . Observe that the last integral in (6.5) can be rewritten as

(6.6) $$ \begin{align} \begin{split} \int_{\mathbb{S}^{n-1}_-}&\int_{\mathbb{S}^{n-1}_-} |w_n| |z_n| \beta(\min\{|w_n|,|z_n|\})\\ &\qquad\quad \,\mathrm{d} S(\bar{L} [j-i],B_H^{n-1}[n-1-j+i],w) \,\mathrm{d} S(\bar{K} [i],B_H^{n-1}[n-1-i],z)\\ &=\int_{\mathbb{S}^{n-1}_+}\int_{\mathbb{S}^{n-1}_+} |w_n| |z_n| \beta(\min\{|w_n|,|z_n|\})\\ &\qquad\quad \,\mathrm{d} S(L [j-i],B_H^{n-1}[n-1-j+i],w) \,\mathrm{d} S(K [i],B_H^{n-1}[n-1-i],z) \end{split} \end{align} $$

for $0\leq i\leq j$ . Similar to the above, we obtain

(6.7) $$ \begin{align} \begin{split} &\int_{\operatorname{SO}(n-1)} \int_{\mathbb{S}^{n-1}}\tilde\beta(|z_n|) \,\mathrm{d} S((K+\vartheta \bar{L}) [j],B_H^{n-1}[n-1-j],z) \,\mathrm{d}\vartheta\\ &\quad= \frac{1}{\kappa_{n-1}} \sum_{i=0}^j \binom{j}{i} \int_{\mathbb{S}^{n-1}_+}\int_{\mathbb{S}^{n-1}_-} |w_n| |z_n| \beta(\min\{|w_n|,|z_n|\})\\ &\quad\qquad\quad \,\mathrm{d} S(L [j-i],B_H^{n-1}[n-1-j+i],w) \,\mathrm{d} S(K [i],B_H^{n-1}[n-1-i],z)\\ &\quad\quad +\frac{1}{\kappa_{n-1}} \sum_{i=0}^j \binom{j}{i} \int_{\mathbb{S}^{n-1}_-}\int_{\mathbb{S}^{n-1}_+} |w_n| |z_n| \beta(\min\{|w_n|,|z_n|\})\\ &\quad\qquad\quad \,\mathrm{d} S(L [j-i],B_H^{n-1}[n-1-j+i],w) \,\mathrm{d} S(K [i],B_H^{n-1}[n-1-i],z). \end{split} \end{align} $$

Thus, combining (6.5), (6.6), and (6.7), we obtain

(6.8) $$ \begin{align} \begin{aligned} &\int_{\operatorname{SO}(n-1)\times \operatorname{O}(1)} \int_{\mathbb{S}^{n-1}} \tilde\beta(|z_n|) \,\mathrm{d} S((K+\eta L) [j],B_H^{n-1}[n-1-j],z) \,\mathrm{d}\eta\\ &\quad= \frac 12 \int_{\operatorname{SO}(n-1)} \int_{\mathbb{S}^{n-1}} \tilde\beta(|z_n|) \,\mathrm{d} S((K+\vartheta L) [j],B_H^{n-1}[n-1-j],z) \,\mathrm{d}\vartheta\\ &\quad\quad+ \frac 12 \int_{\operatorname{SO}(n-1)} \int_{\mathbb{S}^{n-1}} \tilde\beta(|z_n|) \,\mathrm{d} S((K+\vartheta \bar{L}) [j],B_H^{n-1}[n-1-j],z) \,\mathrm{d}\vartheta\\ &\quad= \frac{1}{2\kappa_{n-1}} \sum_{i=0}^j \binom{j}{i} \int_{{\mathbb{S}^{n-1}}}\int_{{\mathbb{S}^{n-1}}} |w_n| |z_n| \beta(\min\{|w_n|,|z_n|\})\\ &\quad\qquad\quad \,\mathrm{d} S(L [j-i],B_H^{n-1}[n-1-j+i],w) \,\mathrm{d} S(K [i],B_H^{n-1}[n-1-i],z), \end{aligned} \end{align} $$

where we used that $|w_n| |z_n| \beta (\min \{|w_n|,|z_n|\})=0$ if $z\in {\mathbb {S}^{n-1}} \cap e_n^{\perp }$ or $w\in {\mathbb {S}^{n-1}} \cap e_n^{\perp }$ . This concludes the proof.

Arguing similarly as in the proof of Corollary 4.3, we obtain from Theorem 1.5 the following equivalent version.

Corollary 6.2 Let $n\geq 2$ . If $0\leq i\leq j\leq n-1$ and $\beta \colon [0,1]\to [0,\infty )$ is measurable, then

$$ \begin{align*} &\int_{\operatorname{SO}(n-1)\times \operatorname{O}(1)} \int_{{\mathbb{S}^{n-1}}} |z_n| \beta(|z_n|) \,\mathrm{d} S(K_1,\ldots,K_i,\eta L_1,\ldots,\eta L_{j-i}),B_H^{n-1}[n-1-j],z) \,\mathrm{d}\eta\\ &\quad= \frac{1}{2\kappa_{n-1}} \int_{{\mathbb{S}^{n-1}}}\int_{{\mathbb{S}^{n-1}}} |w_n| |z_n| \beta(\min\{|w_n|,|z_n|\}) \,\mathrm{d} S(K_1,\ldots,K_i,B_H^{n-1}[n-1-i],z)\\ &\qquad\qquad \,\mathrm{d} S(L_1,\ldots,L_{j-i},B_H^{n-1}[n-1-j+i],w) \end{align*} $$

for $K_1,\ldots ,K_i,L_1,\ldots ,L_{j-i}\in {\mathcal K}^n$ .