Some Obstructions to Contraction Theorems on the Half-Sphere
Abstract
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.
Fields of Study & Subjects
Stochastic processes and statistical mechanics
Mathematical Dynamics and Fractals
Point processes and geometric inequalities
Publication details
- Journal / source
- Progress in probability
- Publisher
- Springer Nature Switzerland
- Pages
- 149–159
- Publication date
- 1 January 2026
- Language
- EN