  In this paper we give a new proof of the (strong) displacement convexity of a
class of integral functionals defined on a compact Riemannian manifold
satisfying a lower Ricci curvature bound. Our approach does not rely on
existence and regularity results for optimal transport maps on Riemannian
manifolds, but it is based on the Eulerian point of view recently introduced by
Otto-Westdickenberg and on the metric characterization of the gradient flows
generated by the functionals in the Wasserstein space.
