{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:ZARMOWN3X2XEWFNZHKWTTKUNWM","short_pith_number":"pith:ZARMOWN3","schema_version":"1.0","canonical_sha256":"c822c759bbbeae4b15b93aad39aa8db328905e313f95bc6d3d6f2783a62e92b0","source":{"kind":"arxiv","id":"2607.12072","version":1},"attestation_state":"computed","paper":{"title":"The maximal volume of projections of the cross-polytope","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Grigory Ivanov","submitted_at":"2026-07-13T18:45:52Z","abstract_excerpt":"We prove the conjectured sharp upper bound for the volume of an arbitrary lower-dimensional orthogonal projection of the regular cross-polytope. More generally, for every spanning family $v_1,\\dots,v_n \\in \\mathbb{R}^k, $ we prove \\[\n  \\operatorname{vol}\\nolimits_{k} \\operatorname{conv} \\{\\pm v_1, \\dots, \\pm v_n\\}\n  \\le \\frac{2^k}{k!}\n  \\sqrt{\\det\\!\\left(\\sum_{i=1}^n v_i\\otimes v_i\\right)}. \\] After the natural normalization, equality holds precisely when the non-zero vectors form an orthonormal basis. We triangulate the boundary of the absolute convex hull, compare the determinant of every ra"},"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":"2607.12072","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2026-07-13T18:45:52Z","cross_cats_sorted":[],"title_canon_sha256":"113cd38aa20a95456487fe10b7fc7e3511245b8860b2591b4b2bff64fb91e67f","abstract_canon_sha256":"28f9a6c45102c0910b19fd88db5808e48438ec9fd8d336c88265648ddcf78564"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-15T00:21:31.662113Z","signature_b64":"zdmaPu/sm+d6SIaQ294zQNOIHR5hrKKRlU9xyN3ODVDOc0WCBIgYmnfkti9rKSyeJF4qVeFQ8joy+hYFlszXAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c822c759bbbeae4b15b93aad39aa8db328905e313f95bc6d3d6f2783a62e92b0","last_reissued_at":"2026-07-15T00:21:31.661282Z","signature_status":"signed_v1","first_computed_at":"2026-07-15T00:21:31.661282Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The maximal volume of projections of the cross-polytope","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Grigory Ivanov","submitted_at":"2026-07-13T18:45:52Z","abstract_excerpt":"We prove the conjectured sharp upper bound for the volume of an arbitrary lower-dimensional orthogonal projection of the regular cross-polytope. More generally, for every spanning family $v_1,\\dots,v_n \\in \\mathbb{R}^k, $ we prove \\[\n  \\operatorname{vol}\\nolimits_{k} \\operatorname{conv} \\{\\pm v_1, \\dots, \\pm v_n\\}\n  \\le \\frac{2^k}{k!}\n  \\sqrt{\\det\\!\\left(\\sum_{i=1}^n v_i\\otimes v_i\\right)}. \\] After the natural normalization, equality holds precisely when the non-zero vectors form an orthonormal basis. We triangulate the boundary of the absolute convex hull, compare the determinant of every ra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.12072","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/2607.12072/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":"2607.12072","created_at":"2026-07-15T00:21:31.661724+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.12072v1","created_at":"2026-07-15T00:21:31.661724+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.12072","created_at":"2026-07-15T00:21:31.661724+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZARMOWN3X2XE","created_at":"2026-07-15T00:21:31.661724+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZARMOWN3X2XEWFNZ","created_at":"2026-07-15T00:21:31.661724+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZARMOWN3","created_at":"2026-07-15T00:21:31.661724+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2608.10241","citing_title":"Geometry of the subgaussian body of an isotropic convex body","ref_index":16,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZARMOWN3X2XEWFNZHKWTTKUNWM","json":"https://pith.science/pith/ZARMOWN3X2XEWFNZHKWTTKUNWM.json","graph_json":"https://pith.science/api/pith-number/ZARMOWN3X2XEWFNZHKWTTKUNWM/graph.json","events_json":"https://pith.science/api/pith-number/ZARMOWN3X2XEWFNZHKWTTKUNWM/events.json","paper":"https://pith.science/paper/ZARMOWN3"},"agent_actions":{"view_html":"https://pith.science/pith/ZARMOWN3X2XEWFNZHKWTTKUNWM","download_json":"https://pith.science/pith/ZARMOWN3X2XEWFNZHKWTTKUNWM.json","view_paper":"https://pith.science/paper/ZARMOWN3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.12072&json=true","fetch_graph":"https://pith.science/api/pith-number/ZARMOWN3X2XEWFNZHKWTTKUNWM/graph.json","fetch_events":"https://pith.science/api/pith-number/ZARMOWN3X2XEWFNZHKWTTKUNWM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZARMOWN3X2XEWFNZHKWTTKUNWM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZARMOWN3X2XEWFNZHKWTTKUNWM/action/storage_attestation","attest_author":"https://pith.science/pith/ZARMOWN3X2XEWFNZHKWTTKUNWM/action/author_attestation","sign_citation":"https://pith.science/pith/ZARMOWN3X2XEWFNZHKWTTKUNWM/action/citation_signature","submit_replication":"https://pith.science/pith/ZARMOWN3X2XEWFNZHKWTTKUNWM/action/replication_record"}},"created_at":"2026-07-15T00:21:31.661724+00:00","updated_at":"2026-07-15T00:21:31.661724+00:00"}