Calculus of Variations and Geometric Measure Theory

G. Carlier - A. Figalli - F. Santambrogio

Transport maps between 1/d-concave densities

created by santambro on 05 Apr 2024


Submitted Paper

Inserted: 5 apr 2024
Last Updated: 5 apr 2024

Year: 2024


In this paper, we extend the scope of Caffarelli's contraction theorem, which provides a measure of the Lipschitz constant for optimal transport maps between log-concave probability densities in $\R^d$. Our focus is on a broader category of densities, specifically those that are $\nicefrac{1}{d}$-concave and can be represented as $V^{-d}$, where $V$ is convex. By setting appropriate conditions, we derive linear or sublinear limitations for the optimal transport map. This leads us to a comprehensive Lipschitz estimate that aligns with the principles established in Caffarelli's theorem.