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Continuity of optimal transport maps and convexity of injectivity domains on the two-sphere

Abstract : 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 of 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 to prove that all injectivity domains are strictly convex too. These results apply for instance on any small C4 -deformation of the two-sphere.
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Alessio Figalli, Ludovic Rifford. Continuity of optimal transport maps and convexity of injectivity domains on the two-sphere. Communications on Pure and Applied Mathematics, Wiley, 2009, 62 (12), pp.1670. ⟨hal-00923261⟩

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