Existence and uniqueness of optimal transport maps

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 m=g·m, for some continuous function g>0, then also (X,d,m) 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.