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On the Poincaré Constant of Log-Concave Measures

  • Patrick CattiauxEmail author
  • Arnaud Guillin
Chapter
  • 52 Downloads
Part of the Lecture Notes in Mathematics book series (LNM, volume 2256)

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.

Keywords

Poincaré inequality Cheeger inequality Log-concave measure Total variation Wasserstein distance Mollification procedure Transference principle 

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Copyright information

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Authors and Affiliations

  1. 1.Institut de Mathématiques de ToulouseUniversité de Toulouse, CNRS UMR 5219ToulouseFrance
  2. 2.Laboratoire de Mathématiques, CNRS UMR 6620Université Clermont-AuvergneAubièreFrance

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