Sectional and intermediate Ricci curvature lower bounds via Optimal Transport

The goal of the paper is to give an optimal transport characterization of lower sectional curvature bounds for smooth $n$-dimensional Riemannian manifolds. More generally we characterize, via optimal transport, 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, $1\leq p\leq n$. Such characterization roughly consists on a convexity condition of the $p$-Reny entropy along $L^{2}$-Wasserstein geodesics, where the role of 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.