  The goal of the paper is to give an optimal transport characterization of
sectional curvature lower (and upper) bounds for smooth $n$-dimensional
Riemannian manifolds. More generally we characterize, via optimal transport,
lower 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 $p$-Renyi 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.
