Optimal transport maps on Alexandrov spaces revisited

Abstract

We give an alternative proof for the fact that in n-dimensional Alexandrov spaces with curvature bounded below there exists a unique optimal transport plan from any purely (n−1)-unrectifiable starting measure, and that this plan is induced by an optimal map. Our proof does not rely on the full optimality of a given plan but rather on the c-monotonicity, thus we obtain the existence of transport maps for wider class of (possibly non-optimal) transport plans.