Nakamura and Tsuji recently obtained an integral inequality involving a Laplace transform of even functions that implies, at the limit, the Blaschke-Santaló inequality in its functional form. Inspired by their method, based on the Fokker-Planck semi-group, we extend the inequality to non-even functions. We consider a well-chosen centering procedure by studying the infimum over translations in a double Laplace transform. This requires a new look on the existing methods and leads to several observations of independent interest on the geometry of the Laplace transform. Application to reverse hypercontractivity is also given.

Let C be a convex body (i.e., a proper closed convex subset with nonempty interior) in a normed space X. We consider four moduli of local uniform rotundity for C at a given point $x\in \partial C$, that extend in a natural way the corresponding notions for the unit ball $B_X$, and we prove that they all coincide. This extends a known result of J. Daneš from 1976 concerning the particular case when $C=B_X$.

Let $H_n$ be the minimal number such that any n-dimensional convex body can be covered by $H_n$ translates of the interior of that body. Similarly $H_n^s$ is the corresponding quantity for symmetric bodies. It is possible to define $H_n$ and $H_n^s$ in terms of illumination of the boundary of the body using external light sources, and the famous Hadwiger’s covering conjecture (illumination conjecture) states that $H_n=H_{n}^s=2^n$. In this note, we obtain new upper bounds on $H_n$ and $H_{n}^s$ for small dimensions n. Our main idea is to cover the body by translates of John’s ellipsoid (the inscribed ellipsoid of the largest volume). Using specific lattice coverings, estimates of quermassintegrals for convex bodies in John’s position, and calculations of mean widths of regular simplexes, we prove the following new upper bounds on $H_n$ and $H_n^s$: $H_5\le 933$, $H_6\le 6137$, $H_7\le 41377$, $H_8\le 284096$, $H_4^s\le 72$, $H_5^s\le 305$, and $H_6^s\le 1292$. For larger n, we describe how the general asymptotic bounds $H_n\le \binom {2n}{n}n(\ln n+\ln \ln n+5)$ and $H_n^s\le 2^n n(\ln n+\ln \ln n+5)$ due to Rogers, Shephard and Roger, Zong, respectively, can be improved for specific values of n.

Gaussian random polytopes have received a lot of attention, especially in the case where the dimension is fixed and the number of points goes to infinity. Our focus is on the less-studied case where the dimension goes to infinity and the number of points is proportional to the dimension d. We study several natural quantities associated with Gaussian random polytopes in this setting. First, we show that the expected number of facets is equal to $C(\alpha)^{d+o(d)}$, where $C(\alpha)$ is some constant which depends on the constant of proportionality $\alpha$. We also extend this result to the expected number of k-facets. We then consider the more difficult problem of the asymptotics of the expected number of pairs of estranged facets of a Gaussian random polytope. When the number of points is 2d, we determine the constant C such that the expected number of pairs of estranged facets is equal to $C^{d+o(d)}$.

A convex body R in the hyperbolic plane is called reduced if any convex body $K\subset R$ has a smaller minimal width than R. We answer a few of Lassak’s questions about ordinary reduced polygons regarding its perimeter, diameter, and circumradius, and we also obtain a hyperbolic extension of a result of Fabińska.

A spectral convex set is a collection of symmetric matrices whose range of eigenvalues forms a symmetric convex set. Spectral convex sets generalize the Schur-Horn orbitopes studied by Sanyal–Sottile–Sturmfels (2011). We study this class of convex bodies, which is closed under intersections, polarity and Minkowski sums. We describe orbits of faces and give a formula for their Steiner polynomials. We then focus on spectral polyhedra. We prove that spectral polyhedra are spectrahedra and give small representations as spectrahedral shadows. We close with observations and questions regarding hyperbolicity cones, polar convex bodies and spectral zonotopes.

Let $\mathbb{P}_\kappa(n)$ be the probability that n points $z_1,\ldots,z_n$ picked uniformly and independently in $\mathfrak{C}_\kappa$, a regular $\kappa$-gon with area 1, are in convex position, that is, form the vertex set of a convex polygon. In this paper, we compute $\mathbb{P}_\kappa(n)$ up to asymptotic equivalence, as $n\to+\infty$, for all $\kappa\geq 3$, which improves on a famous result of Bárány (Ann. Prob. 27, 1999). The second purpose of this paper is to establish a limit theorem which describes the fluctuations around the limit shape of an n-tuple of points in convex position when $n\to+\infty$. Finally, we give an asymptotically exact algorithm for the random generation of $z_1,\ldots,z_n$, conditioned to be in convex position in $\mathfrak{C}_\kappa$.

We prove a functional version of the additive kinematic formula as an application of the Hadwiger theorem on convex functions together with a Kubota-type formula for mixed Monge–Ampère measures. As an application, we give a new explanation for the equivalence of the representations of functional intrinsic volumes as singular Hessian valuations and as integrals with respect to mixed Monge–Ampère measures. In addition, we obtain a new integral geometric formula for mixed area measures of convex bodies, where integration on $\operatorname {SO}(n-1)\times \operatorname {O}(1)$ is considered.

We study a version of the Busemann-Petty problem for $\log $-concave measures with an additional assumption on the dilates of convex, symmetric bodies. One of our main tools is an analog of the classical large deviation principle applied to $\log $-concave measures, depending on the norm of a convex body. We hope this will be of independent interest.

Let $\hat {\mathbb {R}}^n$ be the one-point compactification of $\mathbb {R}^n$ obtained by adding a point at infinity. We say that a subset $A\subseteq \hat {\mathbb {R}}^n$ is $\mathbf {u}$-convex if for every pair of points $\mathbf {z}_1, \mathbf {z}_2 \in A$, the arc of the unique circle through $\mathbf {u}, \mathbf {z}_1$ and $\mathbf {z}_2$, from $\mathbf {z}_1$ to $\mathbf {z}_2$ and not containing $\mathbf {u}$, is contained in A. In this case, we call $\mathbf {u}$ a pole of A. When the pole $\mathbf {u}$ approaches infinity, $\mathbf {u}$-convex sets become convex in the classical sense.

The notion of polar convexity in the complex plane has been used to analyze the behavior of critical points of polynomials. In this paper, we extend the notion to finite-dimensional Euclidean spaces. The goal of this paper is to start building the theory of polar convexity and to show that the introduction of a pole creates a richer theory. For example, polar convexity enjoys a beautiful duality (see Theorem 4.3) that does not exist in classical convexity. We formulate polar analogues of several classical results of the alternatives, such as Gordan’s and Farkas’ lemmas; see Section 5. Finally, we give a full description of the convex hull of finitely many points with respect to finitely many poles; see Theorem 6.7.

In this paper, we consider the discrete Orlicz chord Minkowski problem and solve the existence of this problem, which is the nontrivial extension of the discrete $L_{p}$ chord Minkowski problem for ${0<p<1}$.

Describing the equality conditions of the Alexandrov–Fenchel inequality [Ale37] has been a major open problem for decades. We prove that in the case of convex polytopes, this description is not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level. This is the first hardness result for the problem and is a complexity counterpart of the recent result by Shenfeld and van Handel [SvH23], which gave a geometric characterization of the equality conditions. The proof involves Stanley’s [Sta81] order polytopes and employs poset theoretic technology.

We address the problem of optimal transport with a quadratic cost functional and a constraint on the flux through a constriction along the path. The constriction, conceptually represented by a toll station, limits the flow rate across. We provide a precise formulation which, in addition, is amenable to generalization in higher dimensions. We work out in detail the case of transport in one dimension by proving existence and uniqueness of solution. Under suitable regularity assumptions, we give an explicit construction of the transport plan. Generalization of flux constraints to higher dimensions and possible extensions of the theory are discussed.

In this paper, we give necessary and sufficient conditions for the rigidity of the perimeter inequality under Schwarz symmetrization. The term rigidity refers to the situation in which the equality cases are only obtained by translations of the symmetric set. In particular, we prove that the sufficient conditions for rigidity provided in M. Barchiesi, F. Cagnetti and N. Fusco [Stability of the Steiner symmetrization of convex sets. J. Eur. Math. Soc. 15 (2013), 1245-1278.] are also necessary.

The Hankel index of a real variety X is an invariant that quantifies the difference between nonnegative quadrics and sums of squares on X. In [5], the authors proved an intriguing bound on the Hankel index in terms of the Green–Lazarsfeld index, which measures the ‘linearity’ of the minimal free resolution of the ideal of X. In all previously known cases, this bound was tight. We provide the first class of examples where the bound is not tight; in fact, the difference between Hankel index and Green–Lazarsfeld index can be arbitrarily large. Our examples are outer projections of rational normal curves, where we identify the center of projection with a binary form F. The Green–Lazarsfeld index of the projected curve is given by the complex Waring border rank of F [16]. We show that the Hankel index is given by the almost real rank of F, which is a new notion that comes from decomposing F as a sum of powers of almost real forms. Finally, we determine the range of possible and typical almost real ranks for binary forms.

We establish power-series expansions for the asymptotic expectations of the vertex number and missed area of random disc-polygons in planar convex bodies with $C^{k+1}_+$-smooth boundaries. These results extend asymptotic formulas proved in Fodor et al. (2014).

We present extensions of the colorful Helly theorem for d-collapsible and d-Leray complexes, providing a common generalization to the matroidal versions of the theorem due to Kalai and Meshulam, the ‘very colorful’ Helly theorem introduced by Arocha, Bárány, Bracho, Fabila and Montejano and the ‘semi-intersecting’ colorful Helly theorem proved by Montejano and Karasev.

As an application, we obtain the following extension of Tverberg’s theorem: Let A be a finite set of points in ${\mathbb R}^d$ with $|A|>(r-1)(d+1)$. Then, there exist a partition $A_1,\ldots ,A_r$ of A and a subset $B\subset A$ of size $(r-1)(d+1)$ such that $\cap _{i=1}^r \operatorname {\mathrm {\text {conv}}}( (B\cup \{p\})\cap A_i)\neq \emptyset $ for all $p\in A\setminus B$. That is, we obtain a partition of A into r parts that remains a Tverberg partition even after removing all but one arbitrary point from $A\setminus B$.

Tao and Vu showed that every centrally symmetric convex progression $C\subset \mathbb{Z}^d$ is contained in a generalized arithmetic progression of size $d^{O(d^2)} \# C$. Berg and Henk improved the size bound to $d^{O(d\log d)} \# C$. We obtain the bound $d^{O(d)} \# C$, which is sharp up to the implied constant and is of the same form as the bound in the continuous setting given by John’s theorem.

We prove a weak version of the cross-product conjecture: $\textrm {F}(k+1,\ell ) \hskip .06cm \textrm {F}(k,\ell +1) \ge (\frac 12+\varepsilon ) \hskip .06cm \textrm {F}(k,\ell ) \hskip .06cm \textrm {F}(k+1,\ell +1)$, where $\textrm {F}(k,\ell )$ is the number of linear extensions for which the values at fixed elements $x,y,z$ are k and $\ell $ apart, respectively, and where $\varepsilon>0$ depends on the poset. We also prove the converse inequality and disprove the generalized cross-product conjecture. The proofs use geometric inequalities for mixed volumes and combinatorics of words.

Building on work of Furstenberg and Tzkoni, we introduce $\mathbf {r}$-flag affine quermassintegrals and their dual versions. These quantities generalize affine and dual affine quermassintegrals as averages on flag manifolds (where the Grassmannian can be considered as a special case). We establish affine and linear invariance properties and extend fundamental results to this new setting. In particular, we prove several affine isoperimetric inequalities from convex geometry and their approximate reverse forms. We also introduce functional forms of these quantities and establish corresponding inequalities.