In this talk we present an optimal transport characterization of lower sectional curvature bounds for smooth Riemannian manifolds. More generally, we characterize lower bounds for the $p$-Ricci tensor in terms of convexity of the relative Reny entropy on Wasserstein space with respect to the $p$-dimensional Hausdorff measure. The $p$-Ricci tensor corresponds to taking the trace of the Riemannian curvature tensor on $p$-dimensional planes.

This is a joint work with Andrea Mondino.