Continuity of optimal transport maps and convexity of injectivity domains on small deformations of 𝕊2

Given a compact Riemannian manifold, we study the regularity of the optimal transport 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 condition and to show that this condition, together with the strict convexity on the nonfocal domains, implies the continuity of the optimal transport map. Moreover, our new condition, again combined with the strict convexity of the nonfocal domains, allows us to prove that all injectivity domains are strictly convex too. These results apply, for instance, on any small C⁴‐deformation of the 2‐sphere. © 2009 Wiley Periodicals, Inc.