On the infimum convolution inequality
Volume 189 / 2008
Studia Mathematica 189 (2008), 147-187
MSC: Primary 52A20; Secondary
52A40, 60E15.
DOI: 10.4064/sm189-2-5
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 ${\rm IC}$ inequalities are tied to concentration and study the optimal cost functions for an arbitrary probability measure $\mu$. In particular, we prove an optimal ${\rm IC}$ inequality for product log-concave measures and for uniform measures on the $\ell_p^n$ balls. Such an optimal inequality implies, for a given measure, the central limit theorem of Klartag and the tail estimates of Paouris.
