{"paper":{"title":"The many-body Blaschke-Santal\\'o type inequality via optimal transport","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Dongmeng Xi, Shibing Chen, Yuanyuan Li, Zhe-Feng Xu","submitted_at":"2026-06-29T17:21:26Z","abstract_excerpt":"Let $K_1,\\ldots,K_k\\subset\\mathbb R^n$ be origin-symmetric measurable sets of finite volume such that \\[\n  \\sum_{1\\le i<j\\le k}\\langle x_i,x_j\\rangle\\le \\binom{k}{2},\n  \\qquad \\forall\\,x_i\\in K_i, x_j\\in K_j. \\] We prove the sharp many-body Blaschke--Santal\\'o type inequality \\[\n  \\prod_{i=1}^k |K_i|\\le |B^n|^k \\] proposed by Kalantzopoulos and Saroglou, and characterize all equality cases.\n  The proof combines multi-marginal optimal transport with a pseudo-Euclidean volume estimate. Using the geometric--functional equivalence of Kalantzopoulos and Saroglou, we also establish the functional ve"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.30579","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/2606.30579/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"}