Lower sectional curvature bounds via optimal transport

We give an optimal transport characterization of lower sectional curvature bounds for smooth Riemannian manifolds. More generally we characterize lower (and, in some cases, upper) bounds on the so-called p-Ricci curvature which corresponds to taking the trace of the Riemann curvature tensor on p-dimensional planes. Such characterization roughly consists on a convexity condition of the p-Reny entropy along Wasserstein geodesics, where the role of the reference measure is played by the p-dimensional Hausdorff measure. As application we establish a new Brunn-Minkowski type inequality involving p-dimensional submanifolds and the p-dimensional Hausdorff measure. This is a joint work with Andrea Mondino.