{"paper":{"title":"The non-symmetric Mahler conjecture in dimension three","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"Every convex body in three dimensions has non-symmetric volume product at least 64/9 with respect to its Santaló point.","cross_cats":[],"primary_cat":"math.MG","authors_text":"Dongmeng Xi, Shibing Chen, Yuanyuan Li, Zhefeng Xu","submitted_at":"2026-05-10T05:15:40Z","abstract_excerpt":"We prove the non-symmetric Mahler conjecture in dimension three. More precisely, we prove the sharp lower bound \\[ \\mathcal P(K) \\geq \\frac{64}{9} \\] for every convex body $K \\subset \\mathbb R^3$, where $\\mathcal P(K)$ denotes the non-symmetric volume product with respect to the Santal\\'o point."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove the sharp lower bound P(K) >= 64/9 for every convex body K subset R^3, where P(K) denotes the non-symmetric volume product with respect to the Santaló point.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The claim rests on the existence and uniqueness of the Santaló point for every convex body in R^3 together with the standard definition of the non-symmetric volume product; if either fails for some bodies the bound does not apply.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The non-symmetric Mahler conjecture holds in dimension three: the volume product P(K) satisfies P(K) >= 64/9 for every convex body K in R^3.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Every convex body in three dimensions has non-symmetric volume product at least 64/9 with respect to its Santaló point.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"1563dbb93409d986c954bbeb6341b6d53699b1a8afdc11dd76f46c73570f8914"},"source":{"id":"2605.09334","kind":"arxiv","version":2},"verdict":{"id":"09565008-993a-4db8-9382-e381c748009d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-12T02:41:44.457024Z","strongest_claim":"We prove the sharp lower bound P(K) >= 64/9 for every convex body K subset R^3, where P(K) denotes the non-symmetric volume product with respect to the Santaló point.","one_line_summary":"The non-symmetric Mahler conjecture holds in dimension three: the volume product P(K) satisfies P(K) >= 64/9 for every convex body K in R^3.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The claim rests on the existence and uniqueness of the Santaló point for every convex body in R^3 together with the standard definition of the non-symmetric volume product; if either fails for some bodies the bound does not apply.","pith_extraction_headline":"Every convex body in three dimensions has non-symmetric volume product at least 64/9 with respect to its Santaló point."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.09334/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-20T08:02:04.297470Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T20:33:31.604280Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T13:01:18.610586Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T10:20:43.463555Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"fab45b514bcd770859903e169de0dfbc3b405d8e7c409dd02affc8fa1d072c96"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"9f9f2fb1e387641c62a6c20d15673066eae13d1dca0bde6bfa535eb8556eb14e"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}