Optimal transport maps on Alexandrov spaces revisited

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.