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A proof of the Caffarelli contraction theorem via entropic regularization

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Abstract

We give a new proof of the Caffarelli contraction theorem, which states that the Brenier optimal transport map sending the standard Gaussian measure onto a uniformly log-concave probability measure is Lipschitz. The proof combines a recent variational characterization of Lipschitz transport map by the second author and Juillet with a convexity property of optimizers in the dual formulation of the entropy-regularized optimal transport (or Schrödinger) problem.

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Acknowledgements

MF and NG were supported by ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX- 0002-02. MF and MP were supported by Projects MESA (ANR-18-CE40-006) and EFI (ANR-17-CE40-0030) of the French National Research Agency (ANR). NG is supported by a grant of the Simone and Cino Del Duca Foundation. We thank Christian Léonard for pointing out to us Fortet’s work and sharing a preliminary version of [35] explaining it, Gabriel Peyré for his lectures on numerical optimal transport in March 2019 in Toulouse, and Franck Barthe, Nicolas Juillet and Michel Ledoux for useful discussions. We also thank an anonymous referee for his valuable comments, and for a suggestion that simplified the proof of Theorem 15.

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118, route de Narbonne, 31062, Toulouse Cedex 9, France

    Max Fathi & Maxime Prod’homme

  2. MAP5, UMR 8145, Université Paris Descartes, 45 rue des Saints Pres, 75270, Paris Cedex 06, France

    Nathael Gozlan

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  1. Max Fathi
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  2. Nathael Gozlan
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  3. Maxime Prod’homme
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Correspondence to Max Fathi.

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Communicated by X. Cabre.

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Fathi, M., Gozlan, N. & Prod’homme, M. A proof of the Caffarelli contraction theorem via entropic regularization. Calc. Var. 59, 96 (2020). https://doi.org/10.1007/s00526-020-01754-0

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