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

Eulerian calculus for the displacement convexity in the Wasserstein distance