{"paper":{"title":"A Brenier-Strassen Theorem on CAT(kappa) Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.PR"],"primary_cat":"math.FA","authors_text":"Hugo Malamut (MAP5 - UMR 8145), Nathael Gozlan (MAP5 - UMR 8145), Shin-Ichi Ohta (UOsaka)","submitted_at":"2026-07-29T09:26:03Z","abstract_excerpt":"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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.26671","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.26671/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}