I am interested in a characterization of c-convex/c-concave functions (as in Definition 5.2 and Definition 5.7 in Cédric Villani's book, 'Optimal transport, old and new') on manifolds in the case when the cost function is the half squared distance, $c(x, y) = \frac{1}{2} \text{dist}^2(x, y)$.
In particular, in this comment it is mentioned that on a manifold of nonnegative sectional curvature, a c-convex function $f$ is 1-convex, i.e. $ g(t) = f(\gamma(t)) + \frac{t^2}{2} $ is convex, where $ \gamma $ is a unit speed geodesic. Does anyone know of a reference where this property is explored?
Thank you!