Calculus of Variations and Geometric Measure Theory

S. Daneri - G. Savaré

Eulerian calculus for the displacement convexity in the Wasserstein distance

created by savare on 08 Mar 2008
modified on 12 Jan 2009


Published Paper

Inserted: 8 mar 2008
Last Updated: 12 jan 2009

Journal: SIAM J. Math. Anal.
Volume: 40
Pages: 1104-1122
Year: 2008


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.

Keywords: displacement convexity, Optimal transport, Gradient flows, Ricci curvature