  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.
