ON THE POINCARÉ 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 [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,...).
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https://hal.archives-ouvertes.fr/hal-01898372
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Submitted on : Thursday, October 18, 2018 - 1:14:19 PM
Last modification on : Friday, October 25, 2019 - 1:57:40 AM
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  • HAL Id : hal-01898372, version 1
  • ARXIV : 1810.08369

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Patrick Cattiaux, Arnaud Guillin. ON THE POINCARÉ CONSTANT OF LOG-CONCAVE MEASURES. 2018. ⟨hal-01898372⟩

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