ON THE POINCARÉ CONSTANT OF LOG-CONCAVE MEASURES

Patrick Cattiaux (1) , Arnaud Guillin (2, 3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 LMBP - Laboratoire de Mathématiques Blaise Pascal
3 LMBP - Laboratoire de Mathématiques Blaise Pascal

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 [40] on the link between weak (Poincaré 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 Poincaré (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 Poincaré inequality for the mollified measure. We finally complete the transference section by the comparison of various probability metrics (Fortet-Mourier, bounded-Lipschitz,...).

Mots clés: Poincaré inequality; Cheeger inequality; log-concave measure; total variation; Wasserstein distance; mollification procedure; transference principle

Domaines: Probabilités [math.PR]; Analyse fonctionnelle [math.FA]

Patrick Cattiaux, Arnaud Guillin. ON THE POINCARÉ CONSTANT OF LOG-CONCAVE MEASURES. Geometric Aspects of Functional Analysis, 2256, Springer International Publishing; Springer International Publishing, pp.171-217, 2020, Lecture Notes in Mathematics, ⟨10.1007/978-3-030-36020-7_9⟩. ⟨hal-01898372⟩