On the geometry of metric measure spaces

We introduce and analyze lower (Ricci) curvature bounds Curv ({M, d, m) } ⩾ K for metric measure spaces ({M, d, m) }. Our definition is based on convexity properties of the relative entropy Ent ({ | m. ) } regarded as a function on the L2-Wasserstein space of probability measures on the metric space ({M, d) }. Among others, we show that Curv ({M, d, m) } ⩾ K implies estimates for the volume growth of concentric balls. For Riemannian manifolds, Curv ({M, d, m) } ⩾ K if and only if Ric₌ ({, ) } ⩾ K | |^2 for all TM. The crucial point is that our lower curvature bounds are stable under an appropriate notion of D-convergence of metric measure spaces. We define a complete and separable length metric D on the family of all isomorphism classes of normalized metric measure spaces. The metric D has a natural interpretation, based on the concept of optimal mass transportation. We also prove that the family of normalized metric measure spaces with doubling constant ⩽ C is closed under D-convergence. Moreover, the family of normalized metric measure spaces with doubling constant ⩽ C and diameter ⩽ L is compact under D-convergence.