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On the Poincaré Constant of Log-Concave Measures

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Geometric Aspects of Functional Analysis

Part of the book series: Lecture Notes in Mathematics ((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.

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Author information

Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, Université de Toulouse, CNRS UMR 5219, Toulouse, France

    Patrick Cattiaux

  2. Laboratoire de Mathématiques, CNRS UMR 6620, Université Clermont-Auvergne, Aubière, France

    Arnaud Guillin

Authors
  1. Patrick Cattiaux
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  2. Arnaud Guillin
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Correspondence to Patrick Cattiaux .

Editor information

Editors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel, Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel

    Bo'az Klartag

  2. Department of Mathematics, Technion – Israel Institute of Technology, Haifa, Israel

    Emanuel Milman

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Cattiaux, P., Guillin, A. (2020). On the Poincaré Constant of Log-Concave Measures. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2256. Springer, Cham. https://doi.org/10.1007/978-3-030-36020-7_9

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