Thin-shell concentration for convex measures

TL;DR

This paper demonstrates that s-concave measures with s<0 exhibit thin shell concentration akin to log-concave measures, providing new Berry-Esseen estimates and reverse H"older inequalities for these measures.

Contribution

It extends thin shell concentration results to s-concave measures with s<0 and introduces sharp reverse H"older inequalities for them.

Findings

s-concave measures with s<0 satisfy thin shell concentration

Derived Berry-Esseen type estimates for one-dimensional marginals

Established sharp reverse H"older inequalities for s-concave measures

Abstract

We prove that for $s<0$, $s$-concave measures on ${\mathbb R}^n$ satisfy a thin shell concentration similar to the log-concave one. It leads to a Berry-Esseen type estimate for their one dimensional marginal distributions. We also establish sharp reverse H\"older inequalities for $s$-concave measures.