{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:L4ZA4HTQJ7XHO7T7NRBZGS5XGN","short_pith_number":"pith:L4ZA4HTQ","schema_version":"1.0","canonical_sha256":"5f320e1e704fee777e7f6c43934bb7336f2634e6ce29e67e9f37cb428dcea485","source":{"kind":"arxiv","id":"1603.08850","version":1},"attestation_state":"computed","paper":{"title":"Realizations of Gromov-Hausdorff Distance","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Alexander Ivanov, Alexey Tuzhilin, Stavros Iliadis","submitted_at":"2016-03-29T17:09:22Z","abstract_excerpt":"It is shown that for any two compact metric spaces there exists an \"optimal\" correspondence which the Gromov-Hausdorff distance is attained at. Each such correspondence generates isometric embeddings of these spaces into a compact metric space such that the Gromov-Hausdorff distance between the initial spaces is equal to the Hausdorff distance between their images. Also, the optimal correspondences could be used for constructing the shortest curves in the Gromov-Hausdorff space in exactly the same way as it was done by Alexander Ivanov, Nadezhda Nikolaeva, and Alexey Tuzhilin in arXiv:1504.038"},"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":"1603.08850","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-03-29T17:09:22Z","cross_cats_sorted":[],"title_canon_sha256":"eea062925b72a94cbc0da0a0fad6b745b8dd39b0efe3c00d0700fcb752d9641f","abstract_canon_sha256":"613f5619df1e10dc71e4fc43893671c86dfc911cee7eff05a9a85d30ff75e115"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:18:05.183053Z","signature_b64":"2rtgathFGUg1QkSrfoRm1EwO2K9yLbsa82L24DfTPtZAdz+Nss7WUYDz/o4BDQUQD0Fyh/5tDP5r2v9IYVGkAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5f320e1e704fee777e7f6c43934bb7336f2634e6ce29e67e9f37cb428dcea485","last_reissued_at":"2026-05-18T01:18:05.182403Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:18:05.182403Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Realizations of Gromov-Hausdorff Distance","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Alexander Ivanov, Alexey Tuzhilin, Stavros Iliadis","submitted_at":"2016-03-29T17:09:22Z","abstract_excerpt":"It is shown that for any two compact metric spaces there exists an \"optimal\" correspondence which the Gromov-Hausdorff distance is attained at. Each such correspondence generates isometric embeddings of these spaces into a compact metric space such that the Gromov-Hausdorff distance between the initial spaces is equal to the Hausdorff distance between their images. Also, the optimal correspondences could be used for constructing the shortest curves in the Gromov-Hausdorff space in exactly the same way as it was done by Alexander Ivanov, Nadezhda Nikolaeva, and Alexey Tuzhilin in arXiv:1504.038"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.08850","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":"1603.08850","created_at":"2026-05-18T01:18:05.182513+00:00"},{"alias_kind":"arxiv_version","alias_value":"1603.08850v1","created_at":"2026-05-18T01:18:05.182513+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.08850","created_at":"2026-05-18T01:18:05.182513+00:00"},{"alias_kind":"pith_short_12","alias_value":"L4ZA4HTQJ7XH","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_16","alias_value":"L4ZA4HTQJ7XHO7T7","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_8","alias_value":"L4ZA4HTQ","created_at":"2026-05-18T12:30:29.479603+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2604.11980","citing_title":"Mean dimension of general iterated function systems","ref_index":7,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/L4ZA4HTQJ7XHO7T7NRBZGS5XGN","json":"https://pith.science/pith/L4ZA4HTQJ7XHO7T7NRBZGS5XGN.json","graph_json":"https://pith.science/api/pith-number/L4ZA4HTQJ7XHO7T7NRBZGS5XGN/graph.json","events_json":"https://pith.science/api/pith-number/L4ZA4HTQJ7XHO7T7NRBZGS5XGN/events.json","paper":"https://pith.science/paper/L4ZA4HTQ"},"agent_actions":{"view_html":"https://pith.science/pith/L4ZA4HTQJ7XHO7T7NRBZGS5XGN","download_json":"https://pith.science/pith/L4ZA4HTQJ7XHO7T7NRBZGS5XGN.json","view_paper":"https://pith.science/paper/L4ZA4HTQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1603.08850&json=true","fetch_graph":"https://pith.science/api/pith-number/L4ZA4HTQJ7XHO7T7NRBZGS5XGN/graph.json","fetch_events":"https://pith.science/api/pith-number/L4ZA4HTQJ7XHO7T7NRBZGS5XGN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/L4ZA4HTQJ7XHO7T7NRBZGS5XGN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/L4ZA4HTQJ7XHO7T7NRBZGS5XGN/action/storage_attestation","attest_author":"https://pith.science/pith/L4ZA4HTQJ7XHO7T7NRBZGS5XGN/action/author_attestation","sign_citation":"https://pith.science/pith/L4ZA4HTQJ7XHO7T7NRBZGS5XGN/action/citation_signature","submit_replication":"https://pith.science/pith/L4ZA4HTQJ7XHO7T7NRBZGS5XGN/action/replication_record"}},"created_at":"2026-05-18T01:18:05.182513+00:00","updated_at":"2026-05-18T01:18:05.182513+00:00"}