Related papers: Estimates for moments of general measures on conve…
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…
This article is a survey of recent results on slicing inequalities for convex bodies. The focus is on the setting of arbitrary measures in place of volume.
The aim of this paper is to study properties of sections of convex bodies with respect to different types of measures. We present a formula connecting the Minkowski functional of a convex symmetric body K with the measure of its sections.…
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…
We prove an estimate for arbitrary measure of sections of convex bodies. The proof is based on a stability result for intersection bodies.
A $\sqrt{n}$ estimate in the hyperplane problem with arbitrary measures has recently been proved in \cite{K3}. In this note we present analogs of this result for sections of lower dimensions and in the complex case. We deduce these…
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…
We generalize the hyperplane inequality in dimensions up to 4 to the setting of arbitrary measures in place of the volume. To prove this generalization we establish stability in the affirmative part of the solution to the Busemann-Petty…
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…
A new intrinsic volume metric is introduced for the class of convex bodies in $\mathbb{R}^n$. As an application, an inequality is proved for the asymptotic best approximation of the Euclidean unit ball by arbitrarily positioned polytopes…
We revisit an ingenious argument of K. Ball to provide sharp estimates for the volume of sections of a convex body in John's position. Our technique combines the geometric Brascamp-Lieb inequality with a generalised Parseval-type identity.…
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.…
We prove stability in the affirmative part of the Busemann-Petty problem on sections of complex convex bodies.
We consider a generalization of the hyperplane problem to arbitrary measures in place of volume and to sections of lower dimensions. We prove this generalization for unconditional convex bodies and for duals of bodies with bounded volume…
We propose a new method for obtaining Poincare-type inequalities on arbitrary convex bodies in R^n. Our technique involves a dual version of Bochner's formula and a certain moment map, and it also applies to some non-convex sets. In…
The complex Busemann-Petty problem asks whether origin symmetric convex bodies in C^n with smaller central hyperplane sections necessarily have smaller volume. The answer is affirmative if n\leq 3 and negative if n\geq 4. In this article we…
We introduce complex intersection bodies and show that their properties and applications are similar to those of their real counterparts. In particular, we generalize Busemann's theorem to the complex case by proving that complex…
The generalized Busemann-Petty problem asks whether origin-symmetric convex bodies with lower-dimensional smaller sections necessarily have smaller volume. We study the weighted version of this problem corresponding to the physical…
We prove the following theorem. Let $\mu$ be a measure on $R^n$ with even continuous density, and let $K,L$ be origin-symmetric convex bodies in $R^n$ so that $\mu(K\cap H)\le \mu(L\cap H)$ for any central hyperplane H. Then $\mu(K)\le…
We prove an exponential deviation inequality for the convex hull of a finite sample of i.i.d. random points with a density supported on an arbitrary convex body in $\R^d$, $d\geq 2$. When the density is uniform, our result yields rate…