On the regularity of solutions of optimal transportation problems
 Grégoire Loeper
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We give a necessary and sufficient condition on the cost function so that the map solution of Monge’s optimal transportation problem is continuous for arbitrary smooth positive data. This condition was first introduced by Ma, Trudinger and Wang [24], [30] for a priori estimates of the corresponding Monge–Ampère equation. It is expressed by a socalled costsectional curvature being nonnegative. We show that when the cost function is the squared distance of a Riemannian manifold, the costsectional curvature yields the sectional curvature. As a consequence, if the manifold does not have nonnegative sectional curvature everywhere, the optimal transport map cannot be continuous for arbitrary smooth positive data. The nonnegativity of the costsectional curvature is shown to be equivalent to the connectedness of the contact set between any costconvex function (the proper generalization of a convex function) and any of its supporting functions. When the costsectional curvature is uniformly positive, we obtain that optimal maps are continuous or Hölder continuous under quite weak assumptions on the data, compared to what is needed in the Euclidean case. This case includes the quadratic cost on the round sphere.
 Title
 On the regularity of solutions of optimal transportation problems
 Journal

Acta Mathematica
Volume 202, Issue 2 , pp 241283
 Cover Date
 200906
 DOI
 10.1007/s1151100900378
 Print ISSN
 00015962
 Online ISSN
 18712509
 Publisher
 Springer Netherlands
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 Authors

 Grégoire Loeper ^{(1)}
 Author Affiliations

 1. Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, FR69622, Villeurbanne cedex, France