A R\'enyi entropy interpretation of anti-concentration and noncentral sections of convex bodies

TL;DR

This paper extends concentration bounds to a multivariate entropic framework and provides sharp volume bounds for noncentral sections of convex bodies, linking entropy and geometric properties.

Contribution

It introduces a multivariate entropic approach to concentration inequalities and derives precise volume bounds for noncentral sections of convex bodies, expanding existing geometric and probabilistic methods.

Findings

Extended concentration bounds to multivariate entropic setting

Derived sharp volume bounds for noncentral convex sections

Linked entropy measures with geometric properties of convex bodies

Abstract

We extend Bobkov and Chistyakov's (2015) upper bounds on concentration functions of sums of independent random variables to a multivariate entropic setting. The approach is based on pointwise estimates on densities of sums of independent random vectors uniform on centred Euclidean balls. In this vein, we also obtain sharp bounds on volumes of noncentral sections of isotropic convex bodies.