Inserted: 4 jun 2009
Last Updated: 28 apr 2010
The issue of regularity of optimal transport maps in the case ``cost=squared distance'' on Rn was solved by Caffarelli in the 1990s. However, a major open problem in the theory was the question of regularity for more general cost functions, or for the case ``cost=squared distance'' on a Riemannian manifold. A breakthrough to this problem has been achieved by Ma-Trudinger-Wang and Loeper, who found a necessary and sufficient condition on the cost function in order to ensure the regularity of the optimal map. In the special case ``cost=squared distance'' on a Riemannian manifold, this condition corresponds to the non-negativity of a new curvature tensor on the manifold, which implies strong geometric consequences on the geometry of the manifold and on the structure of its cut-locus.