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Mass Transport and Variants of the Logarithmic Sobolev Inequality

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Abstract

We develop the optimal transportation approach to modified log-Sobolev inequalities and to isoperimetric inequalities. Various sufficient conditions for such inequalities are given. Some of them are new even in the classical log-Sobolev case. The idea behind many of these conditions is that measures with a non-convex potential may enjoy such functional inequalities provided they have a strong integrability property that balances the lack of convexity. In addition, several known criteria are recovered in a simple unified way by transportation methods and generalized to the Riemannian setting.

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, CNRS UMR 5219, Université Paul Sabatier, 31062, Toulouse Cedex 09, France

    Franck Barthe

  2. Moscow State University of Printing Arts, 2A Pryanishnikova, 127550, Moscow, Russia

    Alexander V. Kolesnikov

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  1. Franck Barthe
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  2. Alexander V. Kolesnikov
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Correspondence to Alexander V. Kolesnikov.

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The research of A.V. Kolesnikov was supported by RFBR 07-01-00536, DFG Grant 436 RUS 113/343/0 and GFEN 06-01-39003.

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Barthe, F., Kolesnikov, A.V. Mass Transport and Variants of the Logarithmic Sobolev Inequality. J Geom Anal 18, 921–979 (2008). https://doi.org/10.1007/s12220-008-9039-6

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