A Brenier-Strassen Theorem on CAT(kappa) Spaces

Nathael Gozlan; Hugo Malamut; Shin-Ichi Ohta

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arXiv · 2607.26671

A Brenier-Strassen Theorem on CAT(kappa) Spaces

Nathael Gozlan·Hugo Malamut·Shin-Ichi Ohta

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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 $\mu$, $\nu$ of finite second moment on a complete separable CAT(0) space, we prove that $\mu$ admits a unique W 2 -projection \bar{\mu} to the set of probability measures dominated by $\nu$ in convex order. Moreover, the unique optimal coupling from $\mu$ to \bar{\mu} is induced by a 1-Lipschitz map, without any absolute-continuity assumption on $\mu$. 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 \geq 0, where the optimal map is 1/2-H{\"o}lder continuous. Finally, we give a Strassen-type characterization of one-step barycentric martingales on proper CAT(0) spaces.

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BibTeX RIS

Nathael Gozlan, Hugo Malamut, Shin-Ichi Ohta. 2026-07-29. A Brenier-Strassen Theorem on CAT(kappa) Spaces. https://arxiv.org/abs/2607.26671

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