Continuity and injectivity of optimal maps for non-negatively cross-curved costs

Abstract

Consider transportation of one distribution of mass onto another, chosen to optimize the total expected cost, where cost per unit mass transported from x to y is given by a smooth function c(x,y). If the source density f^+(x) is bounded away from zero and infinity in an open region U' \subset R^n, and the target density f^-(y) is bounded away from zero and infinity on its support clV \subset R^n, which is strongly c-convex with respect to U', and the transportation cost c is non-negatively cross-curved, we deduce continuity and injectivity of the optimal map inside U' (so that the associated potential u belongs to C^1(U')). This result provides a crucial step in the low/interior regularity setting: in a subsequent paper [15], we use it to establish regularity of optimal maps with respect to the Riemannian distance squared on arbitrary products of spheres. The present paper also provides an argument required by Figalli and Loeper to conclude in two dimensions continuity of optimal maps under the weaker (in fact, necessary) hypothesis A3w [17]. In higher dimensions, if the densities f^\pm are H\"older continuous, our result permits continuous differentiability of the map inside U' (in fact, C^{2,\alpha}_{loc} regularity of the associated potential) to be deduced from the work of Liu, Trudinger and Wang [33].