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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"},"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":"2602.20325","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2026-02-23T20:20:57Z","cross_cats_sorted":[],"title_canon_sha256":"f666233eb0ff2ea7b0a2aba42fbe3f1ee7674c00834f5d7f1d024e11053a0d6c","abstract_canon_sha256":"d0abc150e4cf54c80211a39c71704855b970e4ceae1e9d7cb3cc70d237e23927"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-26T01:03:25.970604Z","signature_b64":"a+9P82PzQM34iHY7eEYtYFAX1Aob29ONqtY1qSTuSfH+IUaKAHjVgMz73dGEgtK21EwooE3kPmUT194ZAZZoDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a8ec1ed22ed34aec401dbc77fc779644661ecd0f809d6f66e82262c2e5791e23","last_reissued_at":"2026-05-26T01:03:25.969823Z","signature_status":"signed_v1","first_computed_at":"2026-05-26T01:03:25.969823Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2602.20325","created_at":"2026-05-26T01:03:25.969944+00:00"},{"alias_kind":"arxiv_version","alias_value":"2602.20325v3","created_at":"2026-05-26T01:03:25.969944+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2602.20325","created_at":"2026-05-26T01:03:25.969944+00:00"},{"alias_kind":"pith_short_12","alias_value":"VDWB5URO2NFO","created_at":"2026-05-26T01:03:25.969944+00:00"},{"alias_kind":"pith_short_16","alias_value":"VDWB5URO2NFOYQA5","created_at":"2026-05-26T01:03:25.969944+00:00"},{"alias_kind":"pith_short_8","alias_value":"VDWB5URO","created_at":"2026-05-26T01:03:25.969944+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2604.26828","citing_title":"On the monotonicity of affine quermassintegrals","ref_index":4,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VDWB5URO2NFOYQA5XR37Y54WIR","json":"https://pith.science/pith/VDWB5URO2NFOYQA5XR37Y54WIR.json","graph_json":"https://pith.science/api/pith-number/VDWB5URO2NFOYQA5XR37Y54WIR/graph.json","events_json":"https://pith.science/api/pith-number/VDWB5URO2NFOYQA5XR37Y54WIR/events.json","paper":"https://pith.science/paper/VDWB5URO"},"agent_actions":{"view_html":"https://pith.science/pith/VDWB5URO2NFOYQA5XR37Y54WIR","download_json":"https://pith.science/pith/VDWB5URO2NFOYQA5XR37Y54WIR.json","view_paper":"https://pith.science/paper/VDWB5URO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2602.20325&json=true","fetch_graph":"https://pith.science/api/pith-number/VDWB5URO2NFOYQA5XR37Y54WIR/graph.json","fetch_events":"https://pith.science/api/pith-number/VDWB5URO2NFOYQA5XR37Y54WIR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VDWB5URO2NFOYQA5XR37Y54WIR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VDWB5URO2NFOYQA5XR37Y54WIR/action/storage_attestation","attest_author":"https://pith.science/pith/VDWB5URO2NFOYQA5XR37Y54WIR/action/author_attestation","sign_citation":"https://pith.science/pith/VDWB5URO2NFOYQA5XR37Y54WIR/action/citation_signature","submit_replication":"https://pith.science/pith/VDWB5URO2NFOYQA5XR37Y54WIR/action/replication_record"}},"created_at":"2026-05-26T01:03:25.969944+00:00","updated_at":"2026-05-26T01:03:25.969944+00:00"}