{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:IK4KXDQIATVPH5UKX2KN225LIA","short_pith_number":"pith:IK4KXDQI","schema_version":"1.0","canonical_sha256":"42b8ab8e0804eaf3f68abe94dd6bab40162f881253f96dc18bba517d7b66375b","source":{"kind":"arxiv","id":"2608.07216","version":1},"attestation_state":"computed","paper":{"title":"Geometric Bounds for the Mean Gauge and the Mean Width in Isotropic Position","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Christos Pandis, Silouanos Brazitikos","submitted_at":"2026-08-07T13:34:34Z","abstract_excerpt":"Let $K \\subset \\mathbb{R}^n$ be an origin-symmetric convex body and assume that its uniform probability measure is isotropic in the probabilistic normalization, namely \\[\n  \\int_K x \\otimes x \\, d\\mu_K(x) = \\mathrm{Id}_n. \\] We give deterministic geometric proofs of \\[\n  M(K) \\leq C \\frac{\\log(n)}{\\sqrt{n}}\n  \\qquad \\text{and} \\qquad\n  M^*(K) \\leq C \\sqrt{n} \\, \\log(n), \\] where \\[\n  M(K) = \\int_{\\mathbb{S}^{n-1}} \\|\\theta\\|_K \\, d\\sigma(\\theta),\n  \\qquad\n  M^*(K) = \\int_{\\mathbb{S}^{n-1}} h_K(\\theta) \\, d\\sigma(\\theta). \\] Combining both estimates yields \\[\n  M(K) M^*(K) \\leq C \\log^2(n). \\]\n"},"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":"2608.07216","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2026-08-07T13:34:34Z","cross_cats_sorted":[],"title_canon_sha256":"1a6b994c0a0292b10ad82c4aa0173384ff98ccdb8d422565a2b7953b01287cbf","abstract_canon_sha256":"889103dc2191a78d1c8a81c5ce986818a22f2fa87f6bc1c57a9b51202fd4e00b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-10T01:14:39.888754Z","signature_b64":"gwg6Vz+TrQYZzzrNH5Lk7/vWVop0GCKW0f//Ae/0MrUVAEaMonOQDrX93KOg5YwxgUr/hMfMyVjz9tCeSGyPDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"42b8ab8e0804eaf3f68abe94dd6bab40162f881253f96dc18bba517d7b66375b","last_reissued_at":"2026-08-10T01:14:39.886080Z","signature_status":"signed_v1","first_computed_at":"2026-08-10T01:14:39.886080Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Geometric Bounds for the Mean Gauge and the Mean Width in Isotropic Position","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Christos Pandis, Silouanos Brazitikos","submitted_at":"2026-08-07T13:34:34Z","abstract_excerpt":"Let $K \\subset \\mathbb{R}^n$ be an origin-symmetric convex body and assume that its uniform probability measure is isotropic in the probabilistic normalization, namely \\[\n  \\int_K x \\otimes x \\, d\\mu_K(x) = \\mathrm{Id}_n. \\] We give deterministic geometric proofs of \\[\n  M(K) \\leq C \\frac{\\log(n)}{\\sqrt{n}}\n  \\qquad \\text{and} \\qquad\n  M^*(K) \\leq C \\sqrt{n} \\, \\log(n), \\] where \\[\n  M(K) = \\int_{\\mathbb{S}^{n-1}} \\|\\theta\\|_K \\, d\\sigma(\\theta),\n  \\qquad\n  M^*(K) = \\int_{\\mathbb{S}^{n-1}} h_K(\\theta) \\, d\\sigma(\\theta). \\] Combining both estimates yields \\[\n  M(K) M^*(K) \\leq C \\log^2(n). \\]\n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.07216","kind":"arxiv","version":1},"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/2608.07216/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":"2608.07216","created_at":"2026-08-10T01:14:39.887302+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.07216v1","created_at":"2026-08-10T01:14:39.887302+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.07216","created_at":"2026-08-10T01:14:39.887302+00:00"},{"alias_kind":"pith_short_12","alias_value":"IK4KXDQIATVP","created_at":"2026-08-10T01:14:39.887302+00:00"},{"alias_kind":"pith_short_16","alias_value":"IK4KXDQIATVPH5UK","created_at":"2026-08-10T01:14:39.887302+00:00"},{"alias_kind":"pith_short_8","alias_value":"IK4KXDQI","created_at":"2026-08-10T01:14:39.887302+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/IK4KXDQIATVPH5UKX2KN225LIA","json":"https://pith.science/pith/IK4KXDQIATVPH5UKX2KN225LIA.json","graph_json":"https://pith.science/api/pith-number/IK4KXDQIATVPH5UKX2KN225LIA/graph.json","events_json":"https://pith.science/api/pith-number/IK4KXDQIATVPH5UKX2KN225LIA/events.json","paper":"https://pith.science/paper/IK4KXDQI"},"agent_actions":{"view_html":"https://pith.science/pith/IK4KXDQIATVPH5UKX2KN225LIA","download_json":"https://pith.science/pith/IK4KXDQIATVPH5UKX2KN225LIA.json","view_paper":"https://pith.science/paper/IK4KXDQI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.07216&json=true","fetch_graph":"https://pith.science/api/pith-number/IK4KXDQIATVPH5UKX2KN225LIA/graph.json","fetch_events":"https://pith.science/api/pith-number/IK4KXDQIATVPH5UKX2KN225LIA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IK4KXDQIATVPH5UKX2KN225LIA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IK4KXDQIATVPH5UKX2KN225LIA/action/storage_attestation","attest_author":"https://pith.science/pith/IK4KXDQIATVPH5UKX2KN225LIA/action/author_attestation","sign_citation":"https://pith.science/pith/IK4KXDQIATVPH5UKX2KN225LIA/action/citation_signature","submit_replication":"https://pith.science/pith/IK4KXDQIATVPH5UKX2KN225LIA/action/replication_record"}},"created_at":"2026-08-10T01:14:39.887302+00:00","updated_at":"2026-08-10T01:14:39.887302+00:00"}