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.
Categories: math.AP