Existence and uniqueness of optimal transport maps

Let (X,d,m)(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)(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)(X,d,m) satisfies Assumption 1 and m˜=g⋅m, for some continuous function g>0g>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.