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The goal of the paper is to give an optimal transport characterization ofsectional curvature lower (and upper) bounds for smooth $n$-dimensionalRiemannian manifolds. More generally we characterize, via optimal transport,lower bounds on the so called $p$-Ricci curvature which corresponds to takingthe trace of the Riemann curvature tensor on $p$-dimensional planes, $1\leqp\leq n$. Such characterization roughly consists on a convexity condition ofthe $p$-Renyi entropy along $L^{2}$-Wasserstein geodesics, where the role ofreference measure is played by the $p$-dimensional Hausdorff measure. Asapplication we establish a new Brunn-Minkowski type inequality involving$p$-dimensional submanifolds and the $p$-dimensional Hausdorff measure.

Sectional and intermediate Ricci curvature lower bounds via Optimal Transport