On the infimum convolution inequality

TL;DR

This paper investigates infimum convolution inequalities, exploring their connection to measure concentration, and determines optimal inequalities for various measures, impacting results like the CLT and tail estimates.

Contribution

It establishes the optimal IC-inequalities for product log-concave measures and uniform measures on l_p^n balls, advancing understanding of measure concentration.

Findings

Optimal IC-inequality for product log-concave measures

Optimal IC-inequality for uniform measures on l_p^n balls

Implications for CLT and tail estimates

Abstract

In the paper we study the infimum convolution inequalites. Such an inequality was first introduced by B. Maurey to give the optimal concentration of measure behaviour for the product exponential measure. We show how IC-inequalities are tied to concentration and study the optimal cost functions for an arbitrary probability measure. In particular, we show the optimal IC-inequality for product log-concave measures and for uniform measures on the l_p^n balls. Such an optimal inequality implies, for a given measure, in particular the Central Limit Theorem of Klartag and the tail estimates of Paouris.