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From dimension free concentration to the Poincaré inequality

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Abstract

We prove that a probability measure on an abstract metric space satisfies a non trivial dimension free concentration inequality for the \(\ell _2\) metric if and only if it satisfies the Poincaré inequality. Under some additional assumptions, our result extends to convex sets situation.

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Notes

  1. By locally Lipschitz, we mean Lipschitz on every ball of finite radius.

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Acknowledgments

The authors would like to thank Emanuel Milman for commenting the main result of this paper and for mentioning to them that the method of proof used by Talagrand [41], to prove Corollary 4.1, could be extended to cover general situations.

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Authors and Affiliations

  1. Université Paris Est Marne la Vallée, Laboratoire d’Analyse et de Mathématiques Appliquées (UMR CNRS 8050), 5 bd Descartes, 77454 , Marne la Vallée Cedex 2, France

    Nathael Gozlan & Paul-Marie Samson

  2. Université Paris Ouest Nanterre la Défense, MODAL’X, EA 3454, 200 Avenue de la République, 92000 , Nanterre, France

    Cyril Roberto

Authors
  1. Nathael Gozlan
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  2. Cyril Roberto
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  3. Paul-Marie Samson
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Corresponding author

Correspondence to Nathael Gozlan.

Additional information

Communicated by L. Ambrosio.

The authors were partially supported by the “Agence Nationale de la Recherche” through the grants ANR 2011 BS01 007 01 and ANR 10 LABX-58.

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Gozlan, N., Roberto, C. & Samson, PM. From dimension free concentration to the Poincaré inequality. Calc. Var. 52, 899–925 (2015). https://doi.org/10.1007/s00526-014-0737-6

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