Starting from the pioneering papers of Lott-Villani and Sturm, synthetic and abstract notions of lower Ricci curvature bounds were introduced in the class of complete and separable metric spaces $(X, \sf{d})$ endowed with a locally finite Borel measure $\mathsf m$. This was done prescribing a certain convexity property of an entropy functional along $W_2$-Wasserstein geodesics, leading in this way to the well-known definition of the Curvature Dimension condition $\mathsf{CD(K,N)}$. Among the main properties of such definition, we mention the compatibility with the smooth Riemannian case and the stability with respect to measured Gromov–Hausdorff convergence.
More in general, one can ask the convexity property to hold true on $W_p$-Wasserstein geodesics for any fixed $p>1$, getting to the definition of $\mathsf{CD^p(K,N)}$. In this talk we prove that, under suitable assumptions on the space, the $\mathsf{CD^p(K,N)}$ conditions are all equivalent for every $p>1$. The strategy to do this is to pass through $\mathsf{CD^1}$, a condition that encodes Curvature-Dimension bounds formulated in the language of $L^1$-Optimal Transport.
This is a joint work with A. Akdemir, F. Cavalletti, A. Colinet, R. McCann.