Local semiconvexity of Kantorovich potentials on non-compact manifolds

We prove that any Kantorovich potential for the cost function c = d2/2 on a Riemannian manifold (M, g) 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 μ-measure as soon as the starting measure μ does not charge n – 1-dimensional rectifiable sets