Before revision:  We also show } that relative c-convexity corresponds to the standard notion of convexity in the probability simplex. Hence, we are able to use standard results in optimal transport to establish regularity for the optimal transport maps considered by Pal and Wong. We also provide several new examples of costs satisfying the MTW(0) condition  .
After revision:  We also show } that relative c-convexity geometrically corresponds to geodesic convexity with respect to a dual affine connection. Taken together, these results provide a geometric framework for optimal transport which is complementary to the pseudo-Riemannian theory of Kim and McCann. We provide several applications of this work. In particular, we find a complete K\"ahler surface with non-negative orthogonal bisectional curvature that is not a Hermitian symmetric space or biholomorphic to \mathbb{C .