From concentration to logarithmic Sobolev and Poincaré inequalities

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• 作者：Nathael Gozlan ; ^a ; ^{nathael.gozlan@univ-mlv.fr} ; Cyril Roberto¹ ; ^a ; ^{cyril.roberto@univ-mlv.fr} ; Paul-Marie Samson^a ; ^{paul-marie.samson@univ-mlv.fr}
• 关键词：Concentration of measure ; Transport inequalities ; Poincaré ; inequalities ; Logarithmic-Sobolev inequalities ; Ricci curvature ; Length spaces
• 刊名：Journal of Functional Analysis
• 出版年：2011
• 出版时间：1 March 2011
• 年：2011
• 卷：260
• 期：5
• 页码：1491-1522
• 全文大小：249 K

文摘

We give a new proof of the fact that Gaussian concentration implies the logarithmic Sobolev inequality when the curvature is bounded from below, and also that exponential concentration implies Poincaré inequality under null curvature condition. Our proof holds on non-smooth structures, such as length spaces, and provides a universal control of the constants. We also give a new proof of the equivalence between dimension free Gaussian concentration and Talagrand's transport inequality.