Nonnegative cross-curvature in infinite dimensions: synthetic definition and spaces of measures

TL;DR

This work introduces a synthetic, nonsmooth notion of nonnegative cross-curvature (NNCC) for arbitrary cost spaces, extending the classical MTW framework to infinite dimensions and nonsmooth settings. By using variational c-segments, the authors develop NNCC spaces and prove their stability under products, submersions, and Gromov–Hausdorff limits, and show that Wasserstein spaces inherit NNCC from their base costs. They provide a comprehensive lifting theory, establishing NNCC for Wasserstein costs and giving numerous concrete examples, including Bures–Wasserstein, Fisher–Rao, Hellinger, KL divergence, and Gromov–Wasserstein costs. The paper also clarifies the relationship between NNCC and MTW/Loeper principles, showing that NNCC lifts in some settings but not in others, and highlights applications to gradient flows, functional inequalities, and unbalanced transport. Overall, NNCC offers a robust, scalable geometric framework for understanding curvature-like properties in broad optimal transport settings, with direct implications for infinite-dimensional spaces and measure-valued problems.

Abstract

Nonnegative cross-curvature (NNCC) is a geometric property of a cost function defined on a product space that originates 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.

Figures (1)

• Figure 1: On the left, setting of Example \ref{['ex:MTW_counterexample']} with the four possible c-segments $\mathrm{x}^i$, for $1\le i\le4$, in $(\mathbb{R}^2\times\mathbb{R}^2,c)$ with $c=-\log(|x-y|)$. On the right, plot of the difference of transport costs $f(s)=\mathcal{T}_c(\mu^1(s),\nu)-\mathcal{T}_c(\mu^1(s),\sigma)$, for the lift of c-segments $\mu^1(s)=\frac{1}{2} \delta_{\mathrm{x}_1(s)}+\frac{1}{2} \delta_{\mathrm{x}_3(s)}$, which does not satisfy condition (LMP), since $f(s)>(1-s)f(0)+sf(1)=0$ for every $s\in(0,1)$.

Theorems & Definitions (141)

• Definition 1.1
• Proposition 1.2
• Proposition 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Definition 2.1
• Definition 2.2: MTW tensor
• Definition 2.3: Nonnegative cross-curvature
• Theorem 2.4: kim2012towards, Figalli_Kim_McCann_screening2011