Concentration inequalities for $s$-concave measures of dilations of Borel sets and applications

TL;DR

This paper proves a sharp inequality for dilations of Borel sets under $s$-concave measures, generalizing previous results and deriving dimension-free distribution and Kahane-Khintchine inequalities for functions satisfying Remez type inequalities.

Contribution

It establishes a unified, sharp inequality for $s$-concave measures of dilated sets, extending prior inequalities and applications to function level sets.

Findings

Proved a sharp inequality conjectured by Bobkov.

Generalized Nazarov, Sodin, and Volberg's inequality.

Derived dimension-free distribution and Kahane-Khintchine inequalities.

Abstract

We prove a sharp inequality conjectured by Bobkov on the measure of dilations of Borel sets in $\mathbb{R}^n$ by a $s$-concave probability. Our result gives a common generalization of an inequality of Nazarov, Sodin and Volberg and a concentration inequality of Gu\'edon. Applying our inequality to the level sets of functions satisfying a Remez type inequality, we deduce, as it is classical, that these functions enjoy dimension free distribution inequalities and Kahane-Khintchine type inequalities with positive and negative exponent, with respect to an arbitrary $s$-concave probability.