On the poincar constant of log-concave measures

Abstract

The goal of this paper is to push forward the study of those properties of log-concave measures that help to estimate their Poincaré constant. First we revisit E. Milman’s result (Invent Math 177:1–43, 2009) on the link between weak (Poincaré or concentration) inequalities and Cheeger’s inequality in the log-concave cases, in particular extending localization ideas and a result of Latala, as well as providing a simpler proof of the nice Poincaré (dimensional) bound in the unconditional case. Then we prove alternative transference principle by concentration or using various distances (total variation, Wasserstein). A mollification procedure is also introduced enabling, in the log-concave case, to reduce to the case of the Poincaré inequality for the mollified measure. We finally complete the transference section by the comparison of various probability metrics (Fortet-Mourier, bounded-Lipschitz, …) under a log-concavity assumption.