Displacement convexity of Entropy and the distance cost Optimal Transportation

During the last decade Optimal Transport had a relevant role in the study of geometry of singular spaces that culminated with the Lott–Sturm–Villani theory. The latter is built on the characterisation of Ricci curvature lower bounds in terms of displacement convexity of certain entropy functionals along W2- geodesics. Substantial recent advancements in the theory (localization paradigm and local-to-global property) have been obtained considering the different point of view of L1-Optimal transport problems yielding a different curvature dimension CD1 (K, N) [5] formulated in terms of one-dimensional curvature properties of integral curves of Lipschitz maps. In this note we show that the two approaches produce the same curvature-dimension condition reconciling the two definitions. In particular we show that the CD1 (K, N) condition can be formulated in terms of displacement convexity along W1-geodesics.