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Alexander Koldobsky

Publications and source records attributed to Alexander Koldobsky.

At least 37 records · Page 2Linked to original sources
arXiv2016

Variants of the Busemann-Petty problem and of the Shephard problem

We provide an affirmative answer to a variant of the Busemann-Petty problem, proposed by V.~Milman: Let $K$ be a convex body in ${\mathbb R}^n$ and let $D$ be a compact subset of ${\mathbb R}^n$ such that, for some $1\ls k\ls n-1$, $$|P_F(K)|\ls |D\cap F|$$ for all $F\in G_{n,k}$, where $P_F(K)$ is the orthogonal projection of $K$ onto $F$ and $D\cap F$ is the intersection of $D$ with $F$. Then, $$|K|\ls |D|.$$ We also provide estimates for the lower dimensional Busemann-Petty and Shephard problems, and we prove separation in the original Busemann-Petty problem.

math.MG↗
arXiv2016

A remark on measures of sections of $L_p$-balls

We show that the slicing problem holds true for subspaces of $L_p,p>2$ in the setting of arbitrary measures in place of volume. This generalizes a result of Milman for the original slicing problem.

math.MG↗
arXiv2015

Measures of sections of convex bodies

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.

math.MG↗
arXiv2015

Slicing inequalities for measures of 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 ratio. We also prove it for arbitrary symmetric convex bodies under the condition that the dimension of sections is less than $λn$ for some $λ\in (0,1).$ The constant depends only on $λ.$ Finally, we show that the behavior of the minimal sections for some measures may be different from the case of volume.

math.MG↗
arXiv2014

An isomorphic version of the Busemann-Petty problem for arbitrary measures

We prove the following theorem. Let $μ$ be a measure on $R^n$ with even continuous density, and let $K,L$ be origin-symmetric convex bodies in $R^n$ so that $μ(K\cap H)\le μ(L\cap H)$ for any central hyperplane H. Then $μ(K)\le \sqrt{n} μ(L).$ We also prove this result with better constants for some special classes of measures and bodies. Finally, we prove a version of the hyperplane inequality for convex measures.

math.FA↗
arXiv2013

Estimates for measures of sections of convex 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 inequalities from stability in comparison problems for different generalizations of intersection bodies.

math.MG↗
arXiv2011

Stability and slicing inequalities for intersection bodies

We prove a generalization of the hyperplane inequality for intersection bodies, where volume is replaced by an arbitrary measure $μ$ with even continuous density and sections are of arbitrary dimension $n-k,\ 1\le k 0,\ 1\le k <n.$ Suppose that $K$ and $L$ are origin-symmetric star bodies in $\R^n,$ and $K$ is a generalized $k$-intersection body. If for every $(n-k)$-dimensional subspace $H$ of $\R^n$ $$μ(K\cap H)\leq μ(L\cap H)+\e,$$ then $$μ(K)\leq μ(L) +\frac{n}{n-k}c_{n,k} \vol_n(K)^{k/n}\e.$$

math.MG↗
arXiv2011

A hyperplane inequality for measures of convex bodies in $\R^n,\ n\le 4$

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 problem asking whether symmetric convex bodies in R^n with smaller (n-1)-dimensional volume of all central hyperplane sections necessarily have smaller n-dimensional volume.

math.MG↗
arXiv2011

Stability in the Busemann-Petty and Shephard problems

A comparison problem for volumes of convex bodies asks whether inequalities $f_K(ξ)\le f_L(ξ)$ for all $ξ\in S^{n-1}$ imply that $\vol_n(K)\le \vol_n(L),$ where $K,L$ are convex bodies in $\R^n,$ and $f_K$ is a certain geometric characteristic of $K.$ By linear stability in comparison problems we mean that there exists a constant $c$ such that for every $\e>0$, the inequalities $f_K(ξ)\le f_L(ξ)+\e$ for all $ξ\in S^{n-1}$ imply that $(\vol_n(K))^{\frac{n-1}n}\le (\vol_n(L))^{\frac{n-1}n}+c\e.$ We prove such results in the settings of the Busemann-Petty and Shephard problems and their generalizations. We consider the section function $f_K(ξ)=S_K(ξ)=\vol_{n-1}(K\cap ξ^\bot)$ and the projection function $f_K(ξ)=P_K(ξ)=\vol_{n-1}(K|ξ^\bot),$ where $ξ^\perp$ is the central hyperplane perpendicular to $ξ,$ and $K|ξ^\bot$ is the orthogonal projection of $K$ to $ξ^\bot.$ In these two cases we prove linear stability under additional conditions that $K$ is an intersection body or $L$ is a projection body, respectively. Then we consider other functions $f_K,$ which allows to remove the additional conditions on the bodies in higher dimensions.

math.MG↗
arXiv2009

Positive definite functions and multidimensional versions of random variables

We say that a random vector $X=(X_1,...,X_n)$ in $R^n$ is an $n$-dimensional version of a random variable $Y$ if for any $a\in R^n$ the random variables $\sum a_iX_i$ and $γ(a) Y$ are identically distributed, where $γ:R^n\to [0,\infty)$ is called the standard of $X.$ An old problem is to characterize those functions $γ$ that can appear as the standard of an $n$-dimensional version. In this paper, we prove the conjecture of Lisitsky that every standard must be the norm of a space that embeds in $L_0.$ This result is almost optimal, as the norm of any finite dimensional subspace of $L_p$ with $p\in (0,2]$ is the standard of an $n$-dimensional version ($p$-stable random vector) by the classical result of P.Lèvy. An equivalent formulation is that if a function of the form $f(\|\cdot\|_K)$ is positive definite on $R^n,$ where $K$ is an origin symmetric star body in $R^n$ and $f:R\to R$ is an even continuous function, then either the space $(R^n,\|\cdot\|_K)$ embeds in $L_0$ or $f$ is a constant function. Combined with known facts about embedding in $L_0,$ this result leads to several generalizations of the solution of Schoenberg's problem on positive definite functions.

math.PR↗