{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:7F5OWP4UD2Y7NXQG4SSFNMS64B","short_pith_number":"pith:7F5OWP4U","schema_version":"1.0","canonical_sha256":"f97aeb3f941eb1f6de06e4a456b25ee072a4b8b746c40d35f38092c29edb2dbc","source":{"kind":"arxiv","id":"1709.04976","version":1},"attestation_state":"computed","paper":{"title":"Geometric clustering in normed planes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"cs.CG","authors_text":"Diego Y\\'a\\~nez, Pedro Mart\\'in","submitted_at":"2017-09-13T10:47:10Z","abstract_excerpt":"Given two sets of points $A$ and $B$ in a normed plane, we prove that there are two linearly separable sets $A'$ and $B'$ such that $\\mathrm{diam}(A')\\leq \\mathrm{diam}(A)$, $\\mathrm{diam}(B')\\leq \\mathrm{diam}(B)$, and $A'\\cup B'=A\\cup B.$ This extends a result for the Euclidean distance to symmetric convex distance functions. As a consequence, some Euclidean $k$-clustering algorithms are adapted to normed planes, for instance, those that minimize the maximum, the sum, or the sum of squares of the $k$ cluster diameters. The 2-clustering problem when two different bounds are imposed to the dia"},"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":"1709.04976","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2017-09-13T10:47:10Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"8809d334b8c9828d6ba556eafefd17fc6f563af5bc165811845216b9872923c2","abstract_canon_sha256":"9c18a847782f9063a9d569306cf6995ea899a1e4b43c49a0419440cc0518c880"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:35:08.689593Z","signature_b64":"midg+C8+496c4LwF28dWhAiKASE123oAfqWrF3PTBE7lIMtvn7ADxqavLdCxlBjNdIII0cpyPy6NKf/t3NVMCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f97aeb3f941eb1f6de06e4a456b25ee072a4b8b746c40d35f38092c29edb2dbc","last_reissued_at":"2026-05-18T00:35:08.689164Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:35:08.689164Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Geometric clustering in normed planes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"cs.CG","authors_text":"Diego Y\\'a\\~nez, Pedro Mart\\'in","submitted_at":"2017-09-13T10:47:10Z","abstract_excerpt":"Given two sets of points $A$ and $B$ in a normed plane, we prove that there are two linearly separable sets $A'$ and $B'$ such that $\\mathrm{diam}(A')\\leq \\mathrm{diam}(A)$, $\\mathrm{diam}(B')\\leq \\mathrm{diam}(B)$, and $A'\\cup B'=A\\cup B.$ This extends a result for the Euclidean distance to symmetric convex distance functions. As a consequence, some Euclidean $k$-clustering algorithms are adapted to normed planes, for instance, those that minimize the maximum, the sum, or the sum of squares of the $k$ cluster diameters. The 2-clustering problem when two different bounds are imposed to the dia"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.04976","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":"1709.04976","created_at":"2026-05-18T00:35:08.689219+00:00"},{"alias_kind":"arxiv_version","alias_value":"1709.04976v1","created_at":"2026-05-18T00:35:08.689219+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.04976","created_at":"2026-05-18T00:35:08.689219+00:00"},{"alias_kind":"pith_short_12","alias_value":"7F5OWP4UD2Y7","created_at":"2026-05-18T12:31:05.417338+00:00"},{"alias_kind":"pith_short_16","alias_value":"7F5OWP4UD2Y7NXQG","created_at":"2026-05-18T12:31:05.417338+00:00"},{"alias_kind":"pith_short_8","alias_value":"7F5OWP4U","created_at":"2026-05-18T12:31:05.417338+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/7F5OWP4UD2Y7NXQG4SSFNMS64B","json":"https://pith.science/pith/7F5OWP4UD2Y7NXQG4SSFNMS64B.json","graph_json":"https://pith.science/api/pith-number/7F5OWP4UD2Y7NXQG4SSFNMS64B/graph.json","events_json":"https://pith.science/api/pith-number/7F5OWP4UD2Y7NXQG4SSFNMS64B/events.json","paper":"https://pith.science/paper/7F5OWP4U"},"agent_actions":{"view_html":"https://pith.science/pith/7F5OWP4UD2Y7NXQG4SSFNMS64B","download_json":"https://pith.science/pith/7F5OWP4UD2Y7NXQG4SSFNMS64B.json","view_paper":"https://pith.science/paper/7F5OWP4U","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1709.04976&json=true","fetch_graph":"https://pith.science/api/pith-number/7F5OWP4UD2Y7NXQG4SSFNMS64B/graph.json","fetch_events":"https://pith.science/api/pith-number/7F5OWP4UD2Y7NXQG4SSFNMS64B/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7F5OWP4UD2Y7NXQG4SSFNMS64B/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7F5OWP4UD2Y7NXQG4SSFNMS64B/action/storage_attestation","attest_author":"https://pith.science/pith/7F5OWP4UD2Y7NXQG4SSFNMS64B/action/author_attestation","sign_citation":"https://pith.science/pith/7F5OWP4UD2Y7NXQG4SSFNMS64B/action/citation_signature","submit_replication":"https://pith.science/pith/7F5OWP4UD2Y7NXQG4SSFNMS64B/action/replication_record"}},"created_at":"2026-05-18T00:35:08.689219+00:00","updated_at":"2026-05-18T00:35:08.689219+00:00"}