{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2006:OLCKM4LKRWXNKSXNHPR5CHR5OK","short_pith_number":"pith:OLCKM4LK","schema_version":"1.0","canonical_sha256":"72c4a6716a8daed54aed3be3d11e3d72b15905963d381dc989ec9e38f9be9521","source":{"kind":"arxiv","id":"math/0609447","version":1},"attestation_state":"computed","paper":{"title":"Alexandrov's theorem, weighted Delaunay triangulations, and mixed volumes","license":"","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"math.DG","authors_text":"Alexander I. Bobenko, Ivan Izmestiev","submitted_at":"2006-09-15T15:45:49Z","abstract_excerpt":"We present a constructive proof of Alexandrov's theorem regarding the existence of a convex polytope with a given metric on the boundary. The polytope is obtained as a result of a certain deformation in the class of generalized convex polytopes with the given boundary. We study the space of generalized convex polytopes and discover a relation with the weighted Delaunay triangulations of polyhedral surfaces. The existence of the deformation follows from the non-degeneracy of the Hessian of the total scalar curvature of a positively curved generalized convex polytope. The latter is shown to be e"},"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":"math/0609447","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.DG","submitted_at":"2006-09-15T15:45:49Z","cross_cats_sorted":["math.CO","math.MG"],"title_canon_sha256":"4f97677e16edee2d7898448aacc391cf16e606131002e402bf559ed7ea81be73","abstract_canon_sha256":"3e4aa4579b9721515347044ff6b4283ab81594edb7110b37d6135b5e3150b52c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:36:44.134898Z","signature_b64":"GvsQcdfkddDpb6PdKheG6b0fJbBnazM7R5XrHNgG0nO2Fei+aM5sruezcOaGyNU+UuqucwLvhxKt6LzW/NddCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"72c4a6716a8daed54aed3be3d11e3d72b15905963d381dc989ec9e38f9be9521","last_reissued_at":"2026-05-18T00:36:44.134057Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:36:44.134057Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Alexandrov's theorem, weighted Delaunay triangulations, and mixed volumes","license":"","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"math.DG","authors_text":"Alexander I. Bobenko, Ivan Izmestiev","submitted_at":"2006-09-15T15:45:49Z","abstract_excerpt":"We present a constructive proof of Alexandrov's theorem regarding the existence of a convex polytope with a given metric on the boundary. The polytope is obtained as a result of a certain deformation in the class of generalized convex polytopes with the given boundary. We study the space of generalized convex polytopes and discover a relation with the weighted Delaunay triangulations of polyhedral surfaces. The existence of the deformation follows from the non-degeneracy of the Hessian of the total scalar curvature of a positively curved generalized convex polytope. The latter is shown to be e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0609447","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":""},"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":"math/0609447","created_at":"2026-05-18T00:36:44.134177+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0609447v1","created_at":"2026-05-18T00:36:44.134177+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0609447","created_at":"2026-05-18T00:36:44.134177+00:00"},{"alias_kind":"pith_short_12","alias_value":"OLCKM4LKRWXN","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_16","alias_value":"OLCKM4LKRWXNKSXN","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_8","alias_value":"OLCKM4LK","created_at":"2026-05-18T12:25:54.717736+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/OLCKM4LKRWXNKSXNHPR5CHR5OK","json":"https://pith.science/pith/OLCKM4LKRWXNKSXNHPR5CHR5OK.json","graph_json":"https://pith.science/api/pith-number/OLCKM4LKRWXNKSXNHPR5CHR5OK/graph.json","events_json":"https://pith.science/api/pith-number/OLCKM4LKRWXNKSXNHPR5CHR5OK/events.json","paper":"https://pith.science/paper/OLCKM4LK"},"agent_actions":{"view_html":"https://pith.science/pith/OLCKM4LKRWXNKSXNHPR5CHR5OK","download_json":"https://pith.science/pith/OLCKM4LKRWXNKSXNHPR5CHR5OK.json","view_paper":"https://pith.science/paper/OLCKM4LK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0609447&json=true","fetch_graph":"https://pith.science/api/pith-number/OLCKM4LKRWXNKSXNHPR5CHR5OK/graph.json","fetch_events":"https://pith.science/api/pith-number/OLCKM4LKRWXNKSXNHPR5CHR5OK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OLCKM4LKRWXNKSXNHPR5CHR5OK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OLCKM4LKRWXNKSXNHPR5CHR5OK/action/storage_attestation","attest_author":"https://pith.science/pith/OLCKM4LKRWXNKSXNHPR5CHR5OK/action/author_attestation","sign_citation":"https://pith.science/pith/OLCKM4LKRWXNKSXNHPR5CHR5OK/action/citation_signature","submit_replication":"https://pith.science/pith/OLCKM4LKRWXNKSXNHPR5CHR5OK/action/replication_record"}},"created_at":"2026-05-18T00:36:44.134177+00:00","updated_at":"2026-05-18T00:36:44.134177+00:00"}