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.

中文翻译：

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通过熵正则化证明Caffarelli压缩定理

我们给出了Caffarelli压缩定理的新证明，该定理指出将标准高斯测度发送到一致对数凹概率测度的Brenier最优输运图是Lipschitz。该证明结合了第二作者和Juillet对Lipschitz输运图的最新变化特征以及优化的凸性，在熵规整的最优输运（或Schrödinger）问题的对偶表示中。