Local semiconvexity of Kantorovich potentials on non-compact manifolds

We prove that any Kantorovich potential for the distance-squared cost function on a Riemannian manifold is locally semiconvex in the “region of interest”, without any compactness assumption on M, nor any assumption on its curvature. Such a region of interest is of full \mu-measure as soon as the starting measure \mu does not charge n – 1-dimensional rectifiable sets.