{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:2Y734JH2RRSDMOKOXHJ6P62B25","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":"dc0a57b89c3d2bde75edadcfea74740a361967b7e14081a145b715eb04315a04","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-05-03T21:16:00Z","title_canon_sha256":"0867b203d209d565709f7c5fc1ea3b08ef4b9c7a2df43ab3f2aabb9681171bde"},"schema_version":"1.0","source":{"id":"1605.01094","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.01094","created_at":"2026-05-18T01:15:37Z"},{"alias_kind":"arxiv_version","alias_value":"1605.01094v1","created_at":"2026-05-18T01:15:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.01094","created_at":"2026-05-18T01:15:37Z"},{"alias_kind":"pith_short_12","alias_value":"2Y734JH2RRSD","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"2Y734JH2RRSDMOKO","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"2Y734JH2","created_at":"2026-05-18T12:29:55Z"}],"graph_snapshots":[{"event_id":"sha256:4f92b27e938998f77bf8a8bd9b8e9804a5da2bbb74afb94290562dbb5b707efa","target":"graph","created_at":"2026-05-18T01:15:37Z","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"},"paper":{"abstract_excerpt":"In the present paper we investigate the metric space $\\cal M$ consisting of isometry classes of compact metric spaces, endowed with the Gromov-Hausdorff metric. We show that for any finite subset $M$ from a sufficiently small neighborhood of a generic finite metric space, providing $M$ consists of finite metric spaces with the same number of points, each Steiner minimal tree in $\\cal M$ connecting $M$ is a minimal filling for $M$. As a consequence, we prove that the both Steiner ratio and Gromov-Steiner ratio of $\\cal M$ are equal to $1/2$.","authors_text":"Alexander Ivanov, Alexey Tuzhilin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-05-03T21:16:00Z","title":"Steiner Ratio and Steiner-Gromov Ratio of Gromov-Hausdorff Space"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.01094","kind":"arxiv","version":1},"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:bca1cf6bd9f2cbce3d06e077e76a1a5441068759245addd2b4578b54a7f2d1d5","target":"record","created_at":"2026-05-18T01:15:37Z","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":"dc0a57b89c3d2bde75edadcfea74740a361967b7e14081a145b715eb04315a04","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-05-03T21:16:00Z","title_canon_sha256":"0867b203d209d565709f7c5fc1ea3b08ef4b9c7a2df43ab3f2aabb9681171bde"},"schema_version":"1.0","source":{"id":"1605.01094","kind":"arxiv","version":1}},"canonical_sha256":"d63fbe24fa8c6436394eb9d3e7fb41d76bf953d0551219554c5d59467ab0e745","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d63fbe24fa8c6436394eb9d3e7fb41d76bf953d0551219554c5d59467ab0e745","first_computed_at":"2026-05-18T01:15:37.899963Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:15:37.899963Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wMXhekYlAZrGb957uQ1PCYuiNPoBPBuuf+2XSwrRoiipFAGxx5KwwgEPlzZYSd01Qvio8HfeD2p5P7kui2n0DA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:15:37.900710Z","signed_message":"canonical_sha256_bytes"},"source_id":"1605.01094","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bca1cf6bd9f2cbce3d06e077e76a1a5441068759245addd2b4578b54a7f2d1d5","sha256:4f92b27e938998f77bf8a8bd9b8e9804a5da2bbb74afb94290562dbb5b707efa"],"state_sha256":"929ba74f53ec8597a0329aaae3211895f5fda9588942447e1c90f03f2becbc1c"}