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 Hölder continuous.
