Ricci curvature for metric-measure spaces via optimal transport

摘要: We dene a notion of measured length space X having nonnegative N-Ricci curvature, for N 2 [1;1), or having1-Ricci curvature bounded below byK, forK2 R. The denitions are in terms the displacement convexity certain functions on associated Wasserstein metric P2(X) probability measures. show that these properties preserved under Gromov-Hausdor limits. give geometric and analytic consequences. This paper has dual goals. One goal is to extend results about optimal transport from setting smooth Riemannian manifolds spaces. A second use have Ricci below. refer [11] [44] background material spaces transport, respectively. Further bibliographic notes Appendix F. In present introduction we motivate questions address state main results. To start side, there various reasons try notions more general fairly spaces, meaning (X;d) which distance between two points equals inmum lengths curves joining points. rest this assume compact space. Alexandrov gave good \curvature by K", with K real number, geodesic triangles X. case manifold M induced structure, one recovers sectional K. Length behave nicely respect GromovHausdor topology (modulo isometries); they form