Inserted: 28 jan 2018
Last Updated: 28 mar 2019
Journal: Nonlinear Analysis
Pushing a little forward an approach proposed by Villani, we are going to prove that in the Riemannian setting the condition $\nabla^2 f<g$ implies that $f$ is $c$-concave with respect to the quadratic cost as soon as it has a sufficiently small $C^1$-norm. From this, we deduce a sufficient condition for the optimality of transport maps.
Keywords: Optimal transport, c-concavity, Riemannian manifold, McCann theorem