{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:7HXJXYPY3XP6A6NY3FNJ6PK5M7","short_pith_number":"pith:7HXJXYPY","schema_version":"1.0","canonical_sha256":"f9ee9be1f8dddfe079b8d95a9f3d5d67d0f282b92b21c1f88a74b34814d076f9","source":{"kind":"arxiv","id":"1211.2975","version":1},"attestation_state":"computed","paper":{"title":"On extremums of sums of powered distances to a finite set of points","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Nikolai Nikolov, Rafael Rafailov","submitted_at":"2012-11-13T12:52:52Z","abstract_excerpt":"In this paper we investigate the extremal properties of the sum $$\\sum_{i=1}^n|MA_i|^{\\lambda},$$ where $A_i$ are vertices of a regular simplex, a cross-polytope (orthoplex) or a cube and $M$ varies on a sphere concentric to the sphere circumscribed around one of the given polytopes. We give full characterization for which points on $\\Gamma$ the extremal values of the sum are obtained in terms of $\\lambda$. In the case of the regular dodecahedron and icosahedron in $\\mathbb{R}^3$ we obtain results for which values of $\\lambda$ the corresponding sum is independent of the position of $M$ on $\\Ga"},"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":"1211.2975","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-11-13T12:52:52Z","cross_cats_sorted":[],"title_canon_sha256":"ffabd6df513383c0430b8fc33481e76617ac54e752c5a16b232ac9776a4bd2c5","abstract_canon_sha256":"331b0e2be222b2e45a084329db2304b7e37f1c2c289a9aba9b64b5a837077a3f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:51:23.521194Z","signature_b64":"rqT3TbIbii7MSk6uWpqcMsjw+foGQwN0en67Lrmxq3MK81L6hmquPnWR2EMZpCruir+MwkNBsijnX7KmBTyPBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f9ee9be1f8dddfe079b8d95a9f3d5d67d0f282b92b21c1f88a74b34814d076f9","last_reissued_at":"2026-05-18T02:51:23.520605Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:51:23.520605Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On extremums of sums of powered distances to a finite set of points","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Nikolai Nikolov, Rafael Rafailov","submitted_at":"2012-11-13T12:52:52Z","abstract_excerpt":"In this paper we investigate the extremal properties of the sum $$\\sum_{i=1}^n|MA_i|^{\\lambda},$$ where $A_i$ are vertices of a regular simplex, a cross-polytope (orthoplex) or a cube and $M$ varies on a sphere concentric to the sphere circumscribed around one of the given polytopes. We give full characterization for which points on $\\Gamma$ the extremal values of the sum are obtained in terms of $\\lambda$. In the case of the regular dodecahedron and icosahedron in $\\mathbb{R}^3$ we obtain results for which values of $\\lambda$ the corresponding sum is independent of the position of $M$ on $\\Ga"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.2975","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":"1211.2975","created_at":"2026-05-18T02:51:23.520686+00:00"},{"alias_kind":"arxiv_version","alias_value":"1211.2975v1","created_at":"2026-05-18T02:51:23.520686+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.2975","created_at":"2026-05-18T02:51:23.520686+00:00"},{"alias_kind":"pith_short_12","alias_value":"7HXJXYPY3XP6","created_at":"2026-05-18T12:26:56.085431+00:00"},{"alias_kind":"pith_short_16","alias_value":"7HXJXYPY3XP6A6NY","created_at":"2026-05-18T12:26:56.085431+00:00"},{"alias_kind":"pith_short_8","alias_value":"7HXJXYPY","created_at":"2026-05-18T12:26:56.085431+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/7HXJXYPY3XP6A6NY3FNJ6PK5M7","json":"https://pith.science/pith/7HXJXYPY3XP6A6NY3FNJ6PK5M7.json","graph_json":"https://pith.science/api/pith-number/7HXJXYPY3XP6A6NY3FNJ6PK5M7/graph.json","events_json":"https://pith.science/api/pith-number/7HXJXYPY3XP6A6NY3FNJ6PK5M7/events.json","paper":"https://pith.science/paper/7HXJXYPY"},"agent_actions":{"view_html":"https://pith.science/pith/7HXJXYPY3XP6A6NY3FNJ6PK5M7","download_json":"https://pith.science/pith/7HXJXYPY3XP6A6NY3FNJ6PK5M7.json","view_paper":"https://pith.science/paper/7HXJXYPY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1211.2975&json=true","fetch_graph":"https://pith.science/api/pith-number/7HXJXYPY3XP6A6NY3FNJ6PK5M7/graph.json","fetch_events":"https://pith.science/api/pith-number/7HXJXYPY3XP6A6NY3FNJ6PK5M7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7HXJXYPY3XP6A6NY3FNJ6PK5M7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7HXJXYPY3XP6A6NY3FNJ6PK5M7/action/storage_attestation","attest_author":"https://pith.science/pith/7HXJXYPY3XP6A6NY3FNJ6PK5M7/action/author_attestation","sign_citation":"https://pith.science/pith/7HXJXYPY3XP6A6NY3FNJ6PK5M7/action/citation_signature","submit_replication":"https://pith.science/pith/7HXJXYPY3XP6A6NY3FNJ6PK5M7/action/replication_record"}},"created_at":"2026-05-18T02:51:23.520686+00:00","updated_at":"2026-05-18T02:51:23.520686+00:00"}