A Brenier-Strassen Theorem on CAT(kappa) Spaces

2026/07/29 by Nathael Gozlan, Hugo Malamut, Shin-Ichi Ohta
Mathematics · #math.FA #math.MG #math.PR

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arxiv created 2026/07/29 · arxiv updated 2026/07/30

Abstract

We extend the Brenier-Strassen theorem about projections in convex order to non-flat spaces with curvature bounded from above. Precisely, for probability measures μ, ν of finite second moment on a complete separable CAT(0) space, we prove that μ admits a unique W 2 -projection μ to the set of probability measures dominated by ν in convex order. Moreover, the unique optimal coupling from μ to μ is induced by a 1-Lipschitz map, without any absolute-continuity assumption on μ. Our proof identifies the projection problem with a weak optimal transport problem whose cost is the squared distance to the set of convex means. We also establish a localized version on CAT(kappa) spaces with kappa ≥ 0, where the optimal map is 1/2-Hölder continuous. Finally, we give a Strassen-type characterization of one-step barycentric martingales on proper CAT(0) spaces.

Citations

• Weak Optimal Transport: When is the Dual Potential Convex?
• Wasserstein projections in the convex order: regularity and characterization in the quadratic Gaussian case
• The Fundamental Theorem of Weak Optimal Transport
• Metric extrapolation in the Wasserstein space
• Localisation for constrained transports I: theory
• Backward and Forward Wasserstein Projections in Stochastic Order
• Applications of weak transport theory
• Applications of Strassen's theorem and Choquet theory to optimal transport problems, to uniformly convex functions and to uniformly smooth functions
• A proof of the Caffarelli contraction theorem via entropic regularization
• Weak monotone rearrangement on the line
• Existence, Duality, and Cyclical monotonicity for weak transport costs
• Characterization of a class of weak transport-entropy inequalities on the line
• Kantorovich duality for general transport costs and applications
• The geometry of optimal transportation
• Polar factorization and monotone rearrangement of vector‐valued functions

Related

• Applications of Strassen's theorem and Choquet theory to optimal transport problems, to uniformly convex functions and to uniformly smooth functions
• Existence and uniqueness of optimal transport maps in locally compact CAT(0) spaces
• An optimal transport based characterization of convex order
• Linear extension operators between spaces of Lipschitz maps and optimal transport
• Sampling of probability measures in the convex order by Wasserstein projection