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We prove that if $\Omega\subset \mathbb{R}^{n+1}$ is a (not necessarilystrictly) convex, $C^1$ domain, and $\mu$ and $\bar{\mu}$ are probabilitymeasures 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 acontinuous Monge solution to the optimal transport problem with cost functiongiven by the quadratic distance on the ambient space $\mathbb{R}^{n+1}$. Thisresult is also shown to be sharp, via a counterexample when $\Omega$ isuniformly 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\"oldercontinuous.

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