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.
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/3444
[3] https://www.mpim-bonn.mpg.de/de/node/7138