{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:MOYRO4WPZP3HJAB4SZFCI6Y6P6","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":"d91a8a16a9092915b7263494bcee8f6b5cd699a133f3f8715b91b4bc72e86bbd","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-08-25T09:26:22Z","title_canon_sha256":"9060e973aba6599026a50a287827dd44d87b9594fb56a4ccdc7d33d3b45736fa"},"schema_version":"1.0","source":{"id":"1108.5046","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.5046","created_at":"2026-05-18T04:14:45Z"},{"alias_kind":"arxiv_version","alias_value":"1108.5046v1","created_at":"2026-05-18T04:14:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.5046","created_at":"2026-05-18T04:14:45Z"},{"alias_kind":"pith_short_12","alias_value":"MOYRO4WPZP3H","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_16","alias_value":"MOYRO4WPZP3HJAB4","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_8","alias_value":"MOYRO4WP","created_at":"2026-05-18T12:26:34Z"}],"graph_snapshots":[{"event_id":"sha256:13a5e57f2a737c95d8c6cc7f08741dfcdc2cc640b0bcdf40d0565bb3b1043dbd","target":"graph","created_at":"2026-05-18T04:14:45Z","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":"We give a new proof that a star $\\{op_i:i=1,...,k\\}$ in a normed plane is a Steiner minimal tree of its vertices $\\{o,p_1,...,p_k\\}$ if and only if all angles formed by the edges at o are absorbing [Swanepoel, Networks \\textbf{36} (2000), 104--113]. The proof is more conceptual and simpler than the original one.\n  We also find a new sufficient condition for higher-dimensional normed spaces to share this characterization. In particular, a star $\\{op_i: i=1,...,k\\}$ in any CL-space is a Steiner minimal tree of its vertices $\\{o,p_1,...,p_k\\}$ if and only if all angles are absorbing, which in tur","authors_text":"Horst Martini, Konrad J. Swanepoel, P. Oloff de Wet","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-08-25T09:26:22Z","title":"Absorbing angles, Steiner minimal trees, and antipodality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.5046","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":"sha256:3d16e1367f7207788f24b888a62500400f6824772499d7c3f3e68482480cf06f","target":"record","created_at":"2026-05-18T04:14:45Z","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":"d91a8a16a9092915b7263494bcee8f6b5cd699a133f3f8715b91b4bc72e86bbd","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-08-25T09:26:22Z","title_canon_sha256":"9060e973aba6599026a50a287827dd44d87b9594fb56a4ccdc7d33d3b45736fa"},"schema_version":"1.0","source":{"id":"1108.5046","kind":"arxiv","version":1}},"canonical_sha256":"63b11772cfcbf674803c964a247b1e7fbc780dd72f8163c10e97a78280085cf6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"63b11772cfcbf674803c964a247b1e7fbc780dd72f8163c10e97a78280085cf6","first_computed_at":"2026-05-18T04:14:45.508689Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:14:45.508689Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eHWhZx3g1Qu2mRHSdJK+5sTfQiXS+NnvZ4GdMIxsEBJ0C1imW2aPwOpRZPlskaWxKgz70dNab9r9wdLxbGKiAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:14:45.509119Z","signed_message":"canonical_sha256_bytes"},"source_id":"1108.5046","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3d16e1367f7207788f24b888a62500400f6824772499d7c3f3e68482480cf06f","sha256:13a5e57f2a737c95d8c6cc7f08741dfcdc2cc640b0bcdf40d0565bb3b1043dbd"],"state_sha256":"c0b64b6194b2913d6eeae7339bf9c078c4ddc6e021d1e0da8d4cbc94aaa4cee2"}