Abstract. We define a notion of a measured length space X having nonnegative N -Ricci curvature, for N ∈ [1, ∞), or having ∞-Ricci curvature bounded below by K, for K ∈ R. The definitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P 2 (X) of probability measures. We show that these properties are preserved under measured Gromov-Hausdorff limits. We give geometric and analytic consequences.This paper has dual goals. One goal is to extend results about optimal transport from the setting of smooth Riemannian manifolds to the setting of length spaces. A second goal is to use optimal transport to give a notion for a measured length space to have Ricci curvature bounded below. We refer to [11] and [44] for background material on length spaces and optimal transport, respectively. Further bibliographic notes on optimal transport are in Appendix F. In the present introduction we motivate the questions that we address and we state the main results.To start on the geometric side, there are various reasons to try to extend notions of curvature from smooth Riemannian manifolds to more general spaces. A fairly general setting is that of length spaces, meaning metric spaces (X, d) in which the distance between two points equals the infimum of the lengths of curves joining the points. In the rest of this introduction we assume that X is a compact length space. Alexandrov gave a good notion of a length space having "curvature bounded below by K", with K a real number, in terms of the geodesic triangles in X. In the case of a Riemannian manifold M with the induced length structure, one recovers the Riemannian notion of having sectional curvature bounded below by K. Length spaces with Alexandrov curvature bounded below by K behave nicely with respect to the Gromov-Hausdorff topology on compact metric spaces (modulo isometries); they form a closed subset.In view of Alexandrov's work, it is natural to ask whether there are metric space versions of other types of Riemannian curvature, such as Ricci curvature. This question takes substance from Gromov's precompactness theorem for Riemannian manifolds with Ricci curvature bounded below by K, dimension bounded above by N and diameter bounded above by D [23, Theorem 5.3]. The precompactness indicates that there could be a notion of a length space having "Ricci curvature bounded below by K", special cases of which would be Gromov-Hausdorff limits of manifolds with lower Ricci curvature bounds. Gromov-Hausdorff limits of manifolds with Ricci curvature bounded below have been studied by various authors, notably Cheeger and Colding [15,16,17,18]. One feature of their work, along with the earlier work of Fukaya [21], is that it turns out to be useful to add an auxiliary Borel probability measure ν and consider metric-measure spaces (X, d, ν). (A compact Riemannian manifold M has a canonical measure ν given by the normalized Riemannian density.) There is a measured Gromov-Hausdorff topology on such triples (X, d, ν) (modulo isom...
