Nonnegative Cross-Curvature in Infinite Dimensions: Synthetic Definition and Spaces of Measures

Nonnegative cross-curvature (NNCC) is a geometric property of a cost function defined on a product space originating in optimal transportation and the Ma–Trudinger–Wang theory. Motivated by applications in optimization, gradient flows and mechanism design, we propose a variational formulation of nonnegative cross-curvature on c-convex domains applicable to infinite dimensions and nonsmooth settings. The resulting class of NNCC spaces is closed under Gromov–Hausdorff convergence and for this class, we extend many properties of classical nonnegative cross-curvature: stability under generalized Riemannian submersions, characterization in terms of the convexity of certain sets of c-concave functions, and in the metric case, it is a subclass of positively curved spaces in the sense of Alexandrov. One of our main results is that Wasserstein spaces of probability measures inherit the NNCC property from their base space. Additional examples of NNCC costs include the Bures–Wasserstein and Fisher–Rao squared distances, the Hellinger–Kantorovich squared distance (in some cases), the relative entropy on probability measures, and the 2-Gromov–Wasserstein squared distance on metric measure spaces.