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

Abstract

We prove that if $Ω\subset \mathbb{R}^{n+1}$ is a (not necessarily strictly) convex, $C^1$ domain, and $μ$ and $\barμ$ are probability measures absolutely continuous with respect to surface measure on $\partial Ω$, 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 $Ω$ is uniformly convex but not $C^1$. Additionally, if $Ω$ is $C^{1, α}$ regular for some $α$, then the Monge solution is shown to be Hölder continuous.