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Some obstructions to contraction theorems on the half-sphere

Max Fathi, Matthieu Fradelizi, Nathael Gozlan, Simon Zugmeyer

Abstract

Caffarelli's contraction theorem states that probability measures with uniformly logconcave densities on 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.

Some obstructions to contraction theorems on the half-sphere

Abstract

Caffarelli's contraction theorem states that probability measures with uniformly logconcave densities on 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.
Paper Structure (5 sections, 8 theorems, 32 equations)

This paper contains 5 sections, 8 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Non-contraction of maps from the half-sphere to uniformly-log-concave measures
  3. Optimal transport maps are not $1$-Lipschitz in general
  4. Bijective transport maps are not $1$-Lipschitz in general
  5. Other target domains

Key Result

Theorem 1

Let $\gamma$ be the standard Gaussian measure on $\mathbb{R}^d$, and $\mu$ be a probability measure with density of the form $e^{-V}\gamma$, with $V : \mathbb{R}^n \longrightarrow \mathbb{R}^n \cup\{+\infty\}$ a convex function. Then the Brenier map (or optimal transport map for the quadratic cost)

Theorems & Definitions (13)

  • Theorem 1
  • Proposition 2
  • Remark 3
  • Proposition 4
  • Theorem 5
  • Corollary 6
  • Corollary 7
  • Remark 8
  • Remark 9
  • Proposition 10
  • ...and 3 more