Fabio Cavalletti and Martin Huesmann Self-intersection of optimal geodesics Bulletin of the London Mathematical Society, 46(3): 653-656 2014 http://dx.doi.org/10.1112/blms/bdu026
Abstract: Let (X,d,m)(X,d,m) be a geodesic metric measure space. Consider a geodesic μtμt in the L2L2-Wasserstein space. Then as ss goes to tt, the support of μsμs and the support of μtμt have to overlap, provided an upper bound on the densities holds. We give a more precise formulation of this self-intersection property. Given a geodesic of (X,d,m)(X,d,m) and t∈[0,1]t∈[0,1], we consider the set of times for which this geodesic belongs to the support of μtμt. We prove that tt is a point of Lebesgue density 1 for this set, in the integral sense. Our result applies to spaces satisfying CD(K,∞)CD(K,∞). The non-branching property is not needed.
Abstract: Let (X,d,m) be a proper, non-branching, metric measure space. We show existence and uniqueness of optimal transport maps for cost written as non-decreasing and strictly convex functions of the distance, provided (X,d,m) satisfies a new weak property concerning the behavior of m under the shrinking of sets to points, see Assumption 1. This in particular covers spaces satisfying the measure contraction property.
We also prove a stability property for Assumption 1: If (X,d,m) satisfies Assumption 1 and View the MathML source, for some continuous function g>0, then also View the MathML source verifies Assumption 1. Since these changes in the reference measures do not preserve any Ricci type curvature bounds, this shows that our condition is strictly weaker than measure contraction property.