arXiv subjects

Artem Zvavitch

Publications and source records attributed to Artem Zvavitch.

At least 19 recordsLinked to original sources

arXiv2026

Zonoids whose polars are zonoids: the Banach--Mazur distance need not tend to one

For every $n\geq2$, we consider a Gaussian zonoid of revolution $Z_n$ arising from works of Vitale and Mathis. We prove that $Z_n^*$ is also a zonoid and compute the Banach--Mazur distance from $Z_n$ to the Euclidean ball. This distance is independent of the dimension and is approximately $1.10$. Consequently, the supremal Banach--Mazur distance among zonoids whose polars are zonoids does not converge to $1$. The construction also produces a separable real Banach space $X$, not isometric to a Hilbert space, such that both $X$ and $X^*$ embed linearly isometrically into $L_1$.

math.FA↗

arXiv2026

Simultaneous Busemann-Petty and Shephard Volume Comparisons

We study the simultaneous Busemann-Petty and Shephard volume comparison problem: whether comparison of the volumes of all central hyperplane sections and all orthogonal hyperplane projections determines the ordering of the volumes of two convex bodies. For every $n\geq5$, we construct origin-symmetric convex bodies of revolution $K,L\subset {\mathbb R}^n$ such that every central hyperplane section and every orthogonal hyperplane projection of $K$ has strictly smaller volume than the corresponding section or projection of $L$, while $|K|>|L|$. For $n\leq4$, the affirmative solution of the Busemann--Petty problem shows that the section inequalities alone imply $|K|\leq|L|$. Without origin symmetry, we construct such counterexamples in every dimension $n\geq2$, with one body a nontrivial translate of a Euclidean ball and the other a noncentrally symmetric body of constant brightness.

math.MG↗

arXiv2026

Volume and Projection Inequalities II: Determinants and $L_p$-Sums

We study inequalities for the volume of orthogonal projections and their relation to Firey $L_p$-sum, together with their determinant-power analogues, motivated by the Dembo--Cover--Thomas conjecture. For $L_p$-zonoids $K,L\subset\mathbb{R}^n$ and $u\in S^{n-1}$, we consider the inequality \[ \left( \frac{|K\oplus_p L|} {|P_{u^\perp}(K\oplus_p L)|} \right)^p \geq \left( \frac{|K|}{|P_{u^\perp}K|} \right)^p + \left( \frac{|L|}{|P_{u^\perp}L|} \right)^p . \] For every $1<p<2$, we prove that this inequality fails in every dimension $n\geq2$. In contrast, the weak one-term inequality, obtained by omitting the second term on the right-hand side, holds in dimension two throughout the full range $1\leq p\leq2$. The proof of this planar result uses a sharp estimate for the normalized duality map. We also classify the corresponding determinant-power inequalities in the range $0<p<2$. The strong two-term inequality holds in dimension two and fails in every dimension $n\geq3$. The weak one-term inequality holds for $0<p\leq1$ in dimensions $n\leq3$ and fails for $n\geq4$; for $1<p<2$, it holds only in dimension two.

math.MG↗

arXiv2026

Geometric Large-Deviation-Type Principles for Mixed Measures

We study a geometric analogue of the large deviation principle for mixed measures associated with a class of $\log$-concave probability measures whose densities depend on the gauge of a convex body. For convex bodies in $\mathbb{R}^n$, we prove a geometric large-deviation-type asymptotic for first-order mixed measures, in which the decay under dilation is governed by a natural inradius associated with the measure. In the planar case, we derive an explicit representation and prove a genuine logarithmic limit for second-order mixed measures. As an application, we prove a comparison theorem showing that asymptotic dominance under dilation forces inclusion between convex bodies.

math.PR↗

arXiv2026

Volume and Projection Inequalities I: Zonoids and Courtade's Conjecture

We study volume and projection inequalities for zonoids through the multiaffine determinant polynomials that encode their volumes. We show that a log-submodularity conjecture for the volume of zonoids is equivalent to the Rayleigh property of zonotope volume polynomials, which we prove for the case when the degree or codegree is at most 3. This result is sharp in that when the degree and codegree are at least 4, we construct counterexamples using the existence of non-Rayleigh matroids in ranks at least four. Additionally, we provide unimodular or graphical counterexamples in dimensions four and higher to an equivalent projection inequality formulation of the conjecture. We also show that a stronger projection inequality fails already in dimension three. We next disprove Courtade's conjecture using a pair of orthogonal double bodies of revolution. Although Courtade's conjecture was originally formulated for general convex bodies, we show that it fails even for zonoids in every dimension at least three.

math.MG↗

arXiv2026

Integral inequalities for $α$-convolutions of $α$-concave functions

Classical sumset inequalities originating in additive combinatorics admit geometric analogues for convex bodies in finite-dimensional real vector spaces, as recently developed by Fradelizi and two of the authors. We develop integral analogues for geometric $α$-concave functions under $α$-convolutions, an operation that arises naturally in the ``geometrization of probability'' program. In particular, we establish a Plünnecke--Ruzsa-type inequality, as well as sharp analogues of sum-difference and Ruzsa triangle inequalities, for $α$-convolutions of $α$-concave functions. We also prove a sharp Rogers--Shephard-type inequality and characterize its equality cases for $α$-concave functions, bridging the log-concave case studied by Alonso-Gutiérrez, González-Merino, Jiménez, and Villa and the quasi-concave case studied by Colesanti.

math.FA↗

arXiv2026

Isomorphic Busemann--Petty for arbitrary measures: the sharp order

Let $C_n$ be the optimal constant with the following property. For every even, continuous, strictly positive density $f$ on $R^n$ and all origin-symmetric convex bodies $K,L\subset R^n$, the inequalities $$ \int_{K\capξ^\perp}f \leq \int_{L\capξ^\perp}f \qquad\text{for all }ξ\in S^{n-1} $$ imply $\int_Kf\leq C_n\int_Lf$. In an earlier paper the authors proved that $C_n\leq\sqrt n$. In this paper, we prove the matching lower bound $C_n\geq c\sqrt n$. To simplify the exposition, we first give a complete one-scale construction, based on earlier work of Klartag and Koldobsky, which yields $C_n\geq c\sqrt{n/\log n}$. For the sharp result, we use the random-rounding construction of Klartag and Livshyts as a black box and combine it with a spherical-averaging support-separation argument.

math.FA↗

arXiv2026

On the Fourier Mean Bodies of a Convex Body

In 1998, R. Gardner and G. Zhang introduced the radial $p$th mean bodies $R_pK$ of a convex body $K\subset\mathbb R^n$, $p>-1$, which have since become important objects in geometric tomography. In this paper we study the Fourier transforms of the radial functions of $R_pK$. This leads to a new family of star-shaped sets $F_pK$, which we call the Fourier $p$th mean bodies of $K$. We prove Fourier inversion formulas connecting $R_pK$ and $F_pK$, realizing them as $p$-intersection bodies in the sense of A. Koldobsky. We develop the basic affine geometry of $F_pK$; this includes affine invariance and monotonicity properties. We identify the range of $p$ where $F_p K$ is compact in terms of the decay of $|\widehat{χ_K}|^2$. We show that $F_pK$ is an origin-symmetric convex body for every $0<p\le1$. This range is sharp in general: already for the cube, $F_p[-1,1]^n$ is not convex for $1<p<2$ and $n\geq 2,$ while $F_p[-1,1]^n$ is not compact for $p\geq 2$. We further investigate the features Fourier mean bodies share with intersection bodies: we prove Hensley-type estimates for $F_pK$ when $K$ is isotropic and investigate a few affine isoperimetric inequalities.

math.MG↗

arXiv2026

Weighted Brunn-Minkowski Theory II: Inequalities for Mixed Measures and Applications

In "Weighted Brunn-Minkowski Theory I", the prequel to this work, we discussed how recent developments on concavity of measures have laid the foundations of a nascent weighted Brunn-Minkowski theory. In particular, we defined the mixed measures of three convex bodies and obtained its integral representation. In this work, we obtain inequalities for mixed measures, such as a generalization of Fenchel's inequality; this provides a new, simpler proof of the classical volume case. Moreover, we show that mixed measures are connected to the study of log-submodularity and supermodularity of the measure of Minkowski sums of convex bodies. This elaborates on the recent investigations of these properties for the Lebesgue measure. We conclude by establishing that the only Radon measures that are supermodular over the class of compact, convex sets are multiples of the Lebesgue measure. Motivated by this result, we then discuss weaker forms of supermodularity by restricting the class of convex sets.

math.FA↗

arXiv2024

Measure comparison problems for dilations of convex bodies

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.

math.PR↗

arXiv2024

On the volume of sums of anti-blocking bodies

We study inequalities on the volume of Minkowski sum in the class of anti-blocking bodies. We prove analogues of Plünnecke-Ruzsa type inequality and V. Milman inequality on the concavity of the ratio of volumes of bodies and their projections. We also study Firey $L_p-$sums of anti-blocking bodies and prove Plünnecke-Ruzsa type inequality; V. Milman inequality and Roger-Shephard inequality. The sharp constants are provided in all of those inequalities, for the class of anti-blocking bodies. Finally, we extend our results to the case of unconditional product measures with decreasing density.

math.MG↗

arXiv2023

Comparison Problems for Radon Transforms

Given two non-negative functions $f$ and $g$ such that the Radon transform of $f$ is pointwise smaller than the Radon transform of $g$, does it follow that the $L^p$-norm of $f$ is smaller than the $L^p$-norm of $g$ for a given $p>0$? We consider this problem for the classical and spherical Radon transforms. In both cases we point out classes of functions for which the answer is affirmative, and show that in general the answer is negative if the functions do not belong to these classes. The results are in the spirit of the solution of the Busemann-Petty problem from convex geometry, and the classes of functions that we introduce generalize the class of intersection bodies introduced by Lutwak in 1988. We also deduce slicing inequalities that are related to the well-known Oberlin-Stein type estimates for the Radon transform.

math.FA↗

arXiv2023

Weighted Brunn-Minkowski Theory I: On Weighted Surface Area Measures

The Brunn-Minkowski theory in convex geometry concerns, among other things, the volumes, mixed volumes, and surface area measures of convex bodies. We study generalizations of these concepts to Borel measures with density in $\mathbb{R}^n$-- in particular, the weighted versions of mixed volumes (the so-called mixed measures) when dealing with up to three distinct convex bodies. We then formulate and analyze weighted versions of classical surface area measures, and obtain a new integral formula for the mixed measure of three bodies. As an application, we prove a Bézout-type inequality for rotational invariant log-concave measures, generalizing a result by Artstein-Avidan, Florentin and Ostrover. The results are new and interesting even for the special case of the standard Gaussian measure.

math.MG↗

arXiv2023

Inequalities for sections and projections of convex bodies

This article belongs to the area of geometric tomography, which is the study of geometric properties of solids based on data about their sections and projections. We describe a new direction in geometric tomography where different volumetric results are considered in a more general setting, with volume replaced by an arbitrary measure. Surprisingly, such a general approach works for a number of volumetric results. In particular, we discuss the Busemann-Petty problem on sections of convex bodies for arbitrary measures and the slicing problem for arbitrary measures. We present generalizations of these questions to the case of functions. A number of generalizations of questions related to projections, such as the problem of Shephard, are also discussed as well as some questions in discrete tomography.

math.FA↗

arXiv2023

Volume Product

Our purpose here is to give an overview of known results and open questions concerning the volume product ${\mathcal P}(K)=\min_{z\in K}{\rm vol}(K){\rm vol}((K-z)^*)$ of a convex body $K$ in ${\mathbb R}^n$. We present a number of upper and lower bounds for ${\mathcal P}(K)$, in particular, we discuss the Mahler's conjecture on the lower bound of ${\mathcal P}(K)$, which is still open. We also show connections of ${\mathcal P}(K)$ with different parts of modern mathematics, including Geometric Number Theory, Convex Geometry, Analysis, Harmonic Analysis as well as Systolic and Symplectic Geometries and Probability.

math.MG↗

arXiv2022

General Measure Extensions of Projection Bodies

The inequalities of Petty and Zhang are affine isoperimetric-type inequalities providing sharp bounds for $\text{vol}^{n-1}_{n}(K)\text{vol}_n(Π^\circ K),$ where $ΠK$ is a projection body of a convex body $K$. In this paper, we present a number of generalizations of Zhang's inequality to the setting of arbitrary measures. In addition, we introduce extensions of the projection body operator $Π$ to the setting of arbitrary measures and functions, while providing associated inequalities for this operator; in particular, Zhang-type inequalities. Throughout, we apply shown results to the standard Gaussian measure.

math.FA↗

arXiv2022

On the volume of the Minkowski sum of zonoids

We explore some inequalities in convex geometry restricted to the class of zonoids. We show the equivalence, in the class of zonoids, between a local Alexandrov-Fenchel inequality, a local Loomis-Whitney inequality, the log-submodularity of volume, and the Dembo-Cover-Thomas conjecture on the monotonicity of the ratio of volume to the surface area. In addition to these equivalences, we confirm these conjectures in ${\mathbb R}^3$ and we establish an improved inequality in ${\mathbb R^2}$. Along the way, we give a negative answer to a question of Adam Marcus regarding the roots of the Steiner polynomial of zonoids. We also investigate analogous questions in the $L_p$-Brunn-Minkowski theory, and in particular, we confirm all of the above conjectures in the case $p=2$, in any dimension.

math.MG↗

arXiv2022

Sumset estimates in convex geometry

Sumset estimates, which provide bounds on the cardinality of sumsets of finite sets in a group, form an essential part of the toolkit of additive combinatorics. In recent years, probabilistic or entropic analogs of many of these inequalities were introduced. We study analogues of these sumset estimates in the context of convex geometry and Lebesgue measure on ${\mathbb R}^n$. First, we observe that, with respect to Minkowski summation, volume is supermodular to arbitrary order on the space of convex bodies. Second, we explore sharp constants in the convex geometry analogues of variants of the Plünnecke-Ruzsa inequalities. In the last section of the paper, we provide connections of these inequalities to the classical Rogers-Shephard inequality.

math.MG↗