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Caffarelli’s contraction theorem states that probability measures with uniformly log-concave densities on \(\mathbb {R}^d\) can be realized as the image of a standard Gaussian measure by a globally Lipschitz transport map. We discuss some counterexamples and obstructions that prevent a similar result from holding on the half-sphere endowed with a uniform measure, answering a question of Beck and Jerison.

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Some Obstructions to Contraction Theorems on the Half-Sphere

  • 会议论文

  • 发表日期：

    2026年

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  • Max Fathi,
  • Matthieu Fradelizi,
  • Nathael Gozlan,
  • Simon Zugmeyer

摘要

Caffarelli’s contraction theorem states that probability measures with uniformly log-concave densities on \(\mathbb {R}^d\) can be realized as the image of a standard Gaussian measure by a globally Lipschitz transport map. We discuss some counterexamples and obstructions that prevent a similar result from holding on the half-sphere endowed with a uniform measure, answering a question of Beck and Jerison.