Conditions for existence of single valued optimal transport maps on convex boundaries with nontwisted cost

TL;DR

This paper establishes conditions under which a continuous optimal transport map exists on convex boundaries with certain regularity, extending previous results and providing sharpness and regularity insights.

Contribution

It proves existence of continuous Monge solutions on convex $C^1$ domains with small Wasserstein distance, and shows regularity results including H"older continuity.

Findings

Existence of continuous Monge solutions under specified conditions.

Sharpness of conditions demonstrated via counterexample.

H"older continuity of solutions on $C^{1, eta}$ domains.

Abstract

We prove that if $\Omega\subset \mathbb{R}^{n+1}$ is a (not necessarily strictly) convex, $C^1$ domain, and $\mu$ and $\bar{\mu}$ are probability measures absolutely continuous with respect to surface measure on $\partial \Omega$, with densities bounded away from zero and infinity, whose $2$-Monge-Kantorovich distance is sufficiently small, then there exists a continuous Monge solution to the optimal transport problem with cost function given by the quadratic distance on the ambient space $\mathbb{R}^{n+1}$. This result is also shown to be sharp, via a counterexample when $\Omega$ is uniformly convex but not $C^1$. Additionally, if $\Omega$ is $C^{1, \alpha}$ regular for some $\alpha$, then the Monge solution is shown to be H\"older continuous.