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Using the geometric--functional equivalence of Kalantzopoulos and Saroglou, we also establish the functional ve"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2606.30579","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.AP","submitted_at":"2026-06-29T17:21:26Z","cross_cats_sorted":[],"title_canon_sha256":"1b297262df730563250139641057274d3c9031b436ea2388bdc920328cefc6fc","abstract_canon_sha256":"75481c439c8e8521d8d89234ef1ebc323ec13ac23c477e2191c4f5fc52a425f4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-30T02:18:21.284822Z","signature_b64":"eqJBttZSAflo5YI2MhA0k8CfHwO4Hq1V6A7LxtJCBTD/Zdemf3wRsHNqy1w6fLo9JwYoWP8H9AvN2H4fUJBdDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"563216dfcafc18ed50be4c3fb678d1ecb7ee0d65bb36b252a9132aeba0627aaf","last_reissued_at":"2026-06-30T02:18:21.284304Z","signature_status":"signed_v1","first_computed_at":"2026-06-30T02:18:21.284304Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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