Continuity of optimal transport maps

and convexity of injectivity domains on small deformations of S2 A. Figalli ∗ L. Rifford† Given a compact Riemannian manifold, we study the regularity of the optimal trans-port map between two probability measures with cost given by the squared Riemannian distance. Our strategy is to define a new form of the so-called Ma-Trudinger-Wang con-dition and to show that this condition together with some convexity assumptions on the nonfocal domains implies the continuity of the optimal transport map. Moreover our new condition, again combined with the uniformn convexity of the nonfocal domains, allows to prove that all injectivity domains are strictly convex. These results applies for instance on any small C4-deformation of the two-sphere.