On the infimum convolution inequality

Abstract

We study the infimum convolution inequalities. 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 μ. In particular, we prove an optimal IC inequality for product log-concave measures and for uniform measures on the ℓnp balls. Such an optimal inequality implies, for a given measure, the central limit theorem of Klartag and the tail estimates of Paouris. © Instytut Matematyczny PAN, 2008.