Lipschitz Changes of Variables between Perturbations of Log-concave Measures

TL;DR

This paper extends Caffarelli's result by establishing global Lipschitz transformations between perturbed log-concave measures using optimal transport and Pogorelov estimates, with broader applicability in radially symmetric cases.

Contribution

It introduces a novel approach combining optimal transport and Pogorelov estimates to obtain Lipschitz changes of variables for perturbed log-concave measures, including radially symmetric cases.

Findings

Established global Lipschitz changes for compactly supported perturbations.

Extended Lipschitz change results to radially symmetric measures.

Combined optimal transport with Pogorelov estimates for new bounds.

Abstract

Extending a result of Caffarelli, we provide global Lipschitz changes of variables between compactly supported perturbations of log-concave measures. The result is based on a combination of ideas from optimal transportation theory and a new Pogorelov-type estimate. In the case of radially symmetric measures, Lipschitz changes of variables are obtained for a much broader class of perturbations.