{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:VDWB5URO2NFOYQA5XR37Y54WIR","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"d0abc150e4cf54c80211a39c71704855b970e4ceae1e9d7cb3cc70d237e23927","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2026-02-23T20:20:57Z","title_canon_sha256":"f666233eb0ff2ea7b0a2aba42fbe3f1ee7674c00834f5d7f1d024e11053a0d6c"},"schema_version":"1.0","source":{"id":"2602.20325","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2602.20325","created_at":"2026-05-26T01:03:25Z"},{"alias_kind":"arxiv_version","alias_value":"2602.20325v3","created_at":"2026-05-26T01:03:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2602.20325","created_at":"2026-05-26T01:03:25Z"},{"alias_kind":"pith_short_12","alias_value":"VDWB5URO2NFO","created_at":"2026-05-26T01:03:25Z"},{"alias_kind":"pith_short_16","alias_value":"VDWB5URO2NFOYQA5","created_at":"2026-05-26T01:03:25Z"},{"alias_kind":"pith_short_8","alias_value":"VDWB5URO","created_at":"2026-05-26T01:03:25Z"}],"graph_snapshots":[{"event_id":"sha256:9447188077048ef1d5cc75712a1968baf0f43c5e1c1464b2bf5258ada3112a03","target":"graph","created_at":"2026-05-26T01:03:25Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2602.20325/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Christos Saroglou, K\\'aroly J. B\\\"or\\\"oczky, Konstantinos Patsalos","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2026-02-23T20:20:57Z","title":"On Ball's conjectured Santal\\'o type inequality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.20325","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:44a124aa6426aced3ecd412069112ee04893390edce61b995559cd49fffea36d","target":"record","created_at":"2026-05-26T01:03:25Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"d0abc150e4cf54c80211a39c71704855b970e4ceae1e9d7cb3cc70d237e23927","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2026-02-23T20:20:57Z","title_canon_sha256":"f666233eb0ff2ea7b0a2aba42fbe3f1ee7674c00834f5d7f1d024e11053a0d6c"},"schema_version":"1.0","source":{"id":"2602.20325","kind":"arxiv","version":3}},"canonical_sha256":"a8ec1ed22ed34aec401dbc77fc779644661ecd0f809d6f66e82262c2e5791e23","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a8ec1ed22ed34aec401dbc77fc779644661ecd0f809d6f66e82262c2e5791e23","first_computed_at":"2026-05-26T01:03:25.969823Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-26T01:03:25.969823Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"a+9P82PzQM34iHY7eEYtYFAX1Aob29ONqtY1qSTuSfH+IUaKAHjVgMz73dGEgtK21EwooE3kPmUT194ZAZZoDw==","signature_status":"signed_v1","signed_at":"2026-05-26T01:03:25.970604Z","signed_message":"canonical_sha256_bytes"},"source_id":"2602.20325","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:44a124aa6426aced3ecd412069112ee04893390edce61b995559cd49fffea36d","sha256:9447188077048ef1d5cc75712a1968baf0f43c5e1c1464b2bf5258ada3112a03"],"state_sha256":"7f5ce94c35c5b716b183e8cbd65eefd5ecec3cd26d764c7e9af97d4e8eeb6b9a"}