On the infimum convolution inequality

R. Latała¹ ; J. O. Wojtaszczyk²

¹ Institute of Mathematics University of Warsaw Banacha 2 02-097 Warszawa, Poland and Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 P.O. Box 21 00-956 Warszawa 10, Poland
² Institute of Mathematics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Studia Mathematica, Tome 189 (2008) no. 2, pp. 147-187

Résumé

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.

Détail

DOI : 10.4064/sm189-2-5

Keywords: study infimum convolution inequalities inequality first introduced maurey optimal concentration measure behaviour product exponential measure inequalities tied concentration study optimal cost functions arbitrary probability measure particular prove optimal inequality product log concave measures uniform measures ell balls optimal inequality implies given measure central limit theorem klartag tail estimates paouris