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Alexander Koldobsky

Publications and source records attributed to Alexander Koldobsky.

At least 19 recordsLinked to original sources
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 $$ \int_{K\cap\xi^\perp}f \leq \int_{L\cap\xi^\perp}f \qquad\text{for all }\xi\in S^{n-1} $$ 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.

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arXiv2025

Functions positively associated with integral transforms

We introduce the class of functions positively associated with a linear operator. We describe these classes for several integral operators including the $q$-cosine transform and the spherical Radon transform. We show that positively associated functions control the comparison problem for linear operators generalizing the Busemann-Petty problem for convex bodies.

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arXiv2023

Radon transforms with small derivatives and distance inequalities for convex bodies

Generalizing the slicing inequality for functions on convex bodies from [11], it was proved in [4] that there exists an absolute constant $c$ so that for any $n\in \mathbb N$, any $q\in [0,n-1)$ which is not an odd integer, any origin-symmetric convex body $K$ of volume one in $\mathbb R^n$ and any infinitely smooth probability density $f$ on $K$ we have $$\max_{\xi \in S^{n-1}} {\frac 1{\cos(\pi q/2)}\mathcal R f(\xi, \cdot)_t^{(q)}(0)} \ge \left( \frac {c(q+1)}{n}\right)^{\frac{q+1}2}.$$ Here $\mathcal R f(\xi,t)$ is the Radon transform of $f$, and the fractional derivative of the order $q$ is taken with respect to the variable $t\in \mathbb R$ with fixed $\xi\in S^{n-1}.$ In this note we show that there exist an origin-symmetric convex body $K$ of volume 1 in $\mathbb R^n$ and a continuous probability density $g$ on $K$ so that $$\max_{\xi\in S^{n-1}} {\frac 1{\cos(\pi q/2)}\mathcal R g(\xi, \cdot)_t^{(q)}(0)} \leq \frac 1{\sqrt n} (c(q+1))^{\frac{q+1}2}.$$ In the case $q=0$ this was proved in [5,6], and it was used there to obtain a lower estimate for the maximal outer volume ratio distance from an arbitrary origin-symmetric convex body $K$ to the class of intersection bodies. We extend the latter result to the class $L_{-1-q}^n$ of bodies in $\mathbb R^n$ that embed in $L_{-1-q}.$ Namely, for every $q\in [0,n)$ there exists an origin-symmetric convex body $K$ in $\mathbb R^n$ so that ${d_{\operatorname{ovr}}}(K, L_{-1-q}^n) \ge c n^{\frac 1{2(q+1)}}.$

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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.

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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.

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arXiv2022

An analog of polynomially integrable bodies in even-dimensional spaces

A bounded domain $K \subset \mathbb R^n$ is called polynomially integrable if the $(n-1)$-dimensional volume of the intersection $K$ with a hyperplane $Π$ polynomially depends on the distance from $Π$ to the origin. It was proved in [7] that there are no such domains with smooth boundary if $n$ is even, and if $n$ is odd then the only polynomially integrable domains with smooth boundary are ellipsoids. In this article, we modify the notion of polynomial integrability for even $n$ and consider bodies for which the sectional volume function is a polynomial up to a factor which is the square root of a quadratic polynomial, or, equivalently, the Hilbert transform of this function is a polynomial. We prove that ellipsoids in even dimensions are the only convex infinitely smooth bodies satisfying this property.

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arXiv2021

Inequalities for the Radon transform on convex sets

Several years ago the authors started looking at some problems of convex geometry from a more general point of view, replacing volume by an arbitrary measure. This approach led to new general properties of the Radon transform on convex bodies including an extension of the Busemann-Petty problem and a slicing inequality for arbitrary functions. The latter means that the sup-norm of the Radon transform of any probability density on a convex body of volume one is bounded from below by a positive constant depending only on the dimension. In this note, we prove an inequality that serves as an umbrella for these results

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arXiv2020

Inequalities for the derivatives of the Radon transform on convex bodies

It has been proved that the sup-norm of the Radon transform of an arbitrary probability density on an origin-symmetric convex body of volume 1 is bounded from below by a positive constant depending only on the dimension. In this note we extend this result to the derivatives of the Radon transform. We also prove a comparison theorem for these derivatives.

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arXiv2019

Measure comparison and distance inequalities for convex bodies

We prove new versions of the isomorphic Busemann-Petty problem for two different measures and show how these results can be used to recover slicing and distance inequalities. We also prove a sharp upper estimate for the outer volume ratio distance from an arbitrary convex body to the unit balls of subspaces of $L_p$.

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arXiv2018

On the maximal perimeter of sections of the cube

We prove that the (n-2)-dimensional surface area (perimeter) of central hyperplane sections of the n-dimensional unit cube is maximal for the hyperplane perpendicular to the vector (1,1,0,...,0). This gives a positive answer to a question of Pelczynski who solved the three dimensional case. We study both the real and the complex versions of this problem. We also use our result to show that the answer to an analogue of the Busemann-Petty problem for the surface area is negative in dimensions 14 and higher.

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arXiv2017

Estimates for moments of general measures on convex bodies

We prove several estimates for the moments of arbitrary measures on convex bodies. We apply these estimates to show a new slicing inequality for measures on convex bodies. We also deduce estimates for the outer volume ratio distance from an arbitrary centrally-symmetric convex body in R^n to the class of unit balls of n-dimensional subspaces of L_p-spaces. Finally, we prove a result of the Busemann-Petty type for these moments.

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arXiv2017

Estimating volume and surface area of a convex body via its projections or sections

The main goal of this paper is to present a series of inequalities connecting the surface area measure of a convex body and surface area measure of its projections and sections. We present a solution of a question from S. Campi, P. Gritzmann and P. Gronchi regarding the asymptotic behavior of the best constant in a recently proposed reverse Loomis-Whitney inequality. Next we give a new sufficient condition for the slicing problem to have an affirmative answer, in terms of the least "outer volume ratio distance" from the class of intersection bodies of projections of at least proportional dimension of convex bodies. Finally, we show that certain geometric quantities such as the volume ratio and minimal surface area (after a suitable normalization) are not necessarily close to each other.

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arXiv2017

An example related to the slicing inequality for general measures

For $n\in \mathbb{N}$ let $S_n$ be the smallest number $S>0$ satisfying the inequality $$ \int_K f \le S \cdot |K|^{\frac 1n} \cdot \max_{ξ\in S^{n-1}} \int_{K\cap ξ^\bot} f $$ for all centrally-symmetric convex bodies $K$ in $\mathbb{R}^n$ and all even, continuous probability densities $f$ on $K$. Here $|K|$ is the volume of $K$. It was proved by the second-named author that $S_n\le 2\sqrt{n}$, and in analogy with Bourgain's slicing problem, it was asked whether $S_n$ is bounded from above by a universal constant. In this note we construct an example showing that $S_n\ge c\sqrt{n}/\sqrt{\log \log n},$ where $c > 0$ is an absolute constant. Additionally, for any $0 < α< 2$ we describe a related example that satisfies the so-called $ψ_α$-condition.

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arXiv2017

On polynomially integrable convex bodies

An infinitely smooth convex body in $\mathbb R^n$ is called polynomially integrable of degree $N$ if its parallel section functions are polynomials of degree $N$. We prove that the only smooth convex bodies with this property in odd dimensions are ellipsoids, if $N\ge n-1$. This is in contrast with the case of even dimensions and the case of odd dimensions with $N<n-1$, where such bodies do not exist, as it was recently shown by Agranovsky.

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arXiv2016

Volume difference inequalities

We prove several inequalities estimating the distance between volumes of two bodies in terms of the maximal or minimal difference between areas of sections or projections of these bodies. We also provide extensions in which volume is replaced by an arbitrary measure.

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arXiv2016

On the average volume of sections of convex bodies

The average section functional ${\rm as}(K)$ of a centered convex body in ${\mathbb R}^n$ is the average volume of central hyperplane sections of $K$: \begin{equation*}{\rm as}(K)=\int_{S^{n-1}}|K\cap ξ^{\perp }|\,dσ(ξ).\end{equation*} We study the question if there exists an absolute constant $C>0$ such that for every $n$, for every centered convex body $K$ in ${\mathbb R}^n$ and for every 0<k<n, $${\rm as}(K)\ls C^k|K|^{\frac{k}{n}}\,\max_{E\in {\rm Gr}_{n-k}}{\rm as}(K\cap E).$$ We observe that the case $k=1$ is equivalent to the hyperplane conjecture. We show that this inequality holds true in full generality if one replaces $C$ by $CL_K$ or $Cd_{\rm ovr}(K,{\cal{BP}}_k^n)$, where $L_K$ is the isotropic constant of $K$ and $d_{\rm ovr}(K,{\cal{BP}}_k^n)$ is the outer volume ratio distance from $K$ to the class ${\cal{BP}}_k^n$ of generalized $k$-intersection bodies. We also compare ${\rm as}(K)$ to the average of ${\rm as}(K\cap E)$ over all $k$-codimensional sections of $K$. We examine separately the dependence of the constants on the dimension in the case where $K$ is in some of the classical positions as well as the natural lower dimensional analogue of the average section functional.

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arXiv2016

Inequalities for the surface area of projections of convex bodies

We provide general inequalities that compare the surface area S(K) of a convex body K in ${\mathbb R}^n$ to the minimal, average or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for all the quermassintegrals of K. We examine separately the dependence of the constants on the dimension in the case where K is in some of the classical positions or K is a projection body. Our results are in the spirit of the hyperplane problem, with sections replaced by projections and volume by surface area.

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