On the Poincar{\'e} constant of log-concave measures

TL;DR

This paper advances understanding of the Poincaré constant for log-concave measures by extending inequalities, introducing transference principles, and comparing metrics, with implications for concentration and dimensional bounds.

Contribution

It provides new proofs, extends localization techniques, and develops transference principles for Poincaré constants in log-concave measures.

Findings

Extended Milman's result linking weak inequalities and Cheeger's inequality.

Introduced a mollification procedure to analyze Poincaré inequalities.

Compared various probability metrics for measure transference.

Abstract

The goal of this paper is to push forward the study of those properties of log-concave measures that help to estimate their Poincar{\'e} constant. First we revisit E. Milman's result [40] on the link between weak (Poincar{\'e} or concentration) inequalities and Cheeger's inequality in the logconcave cases, in particular extending localization ideas and a result of Latala, as well as providing a simpler proof of the nice Poincar{\'e} (dimensional) bound in the inconditional 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 logconcave case, to reduce to the case of the Poincar{\'e} inequality for the mollified measure. We finally complete the transference section by the comparison of various probability metrics (Fortet-Mourier, bounded-Lipschitz,...).