{"paper":{"title":"On Ball's conjectured Santal\\'o type inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Christos Saroglou, K\\'aroly J. B\\\"or\\\"oczky, Konstantinos Patsalos","submitted_at":"2026-02-23T20:20:57Z","abstract_excerpt":"We prove that if $K$ is a symmetric and isotropic convex body in $\\mathbb{R}^n$, then $$\\int_K\\langle x,u\\rangle^2\\,dx\\int_{K^\\circ}\\langle x,u\\rangle^2\\,dx\\leq \\left(\\int_{B_2^n}\\langle x,u\\rangle^2\\,dx\\right)^2,\\qquad\\forall u\\in\\mathbb{R}^n,$$with equality for some $u\\neq o$, if and only if $K$ is a Euclidean ball. This confirms a conjecture by Keith Ball (1986), stating that for any symmetric convex body $K$ in $\\mathbb{R}^n$, it holds $$\\int_K\\int_{K^\\circ}\\langle x,y\\rangle^2\\,dx\\,dy\\leq \\int_{B_2^n}\\int_{B_2^n}\\langle x,y\\rangle^2\\,dx\\,dy,$$with equality if and only if $K$ is an ellipso"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.20325","kind":"arxiv","version":3},"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/2602.20325/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"}