Displacement convexity of Entropy and the distance cost Optimal Transportation

Abstract

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 $W_{2}$-geodesics. Substantial recent advancements in the theory (localization paradigm and local-to-global property) have been obtained considering the different point of view of $L^1$-Optimal transport problems yielding a different curvature dimension $\mathsf{CD}^{1}(K,N)$ [8] 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 $\mathsf{CD}^{1}(K,N)$ condition can be formulated in terms of displacement convexity along $W_{1}$-geodesics.