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 RicM ( ξ ,ξ ) ⩾ K| ξ |² for all ξ ∈ 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.