Self-intersection of optimal geodesics

Let (X, d, m) be a geodesic metric measure space. Consider a geodesic mu(t) in the L-2-Wasserstein space. Then as s goes to t, the support of mu(s) and the support of mu(t) have to overlap, provided an upper bound on the densities holds. We give a more precise formulation of this self-intersection property. Given a geodesic of (X, d, m) and t is an element of[0, 1], we consider the set of times for which this geodesic belongs to the support of mu(t). We prove that t is a point of Lebesgue density 1 for this set, in the integral sense. Our result applies to spaces satisfying CD(K, infinity). The non-branching property is not needed.