{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:7F5OWP4UD2Y7NXQG4SSFNMS64B","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":"9c18a847782f9063a9d569306cf6995ea899a1e4b43c49a0419440cc0518c880","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2017-09-13T10:47:10Z","title_canon_sha256":"8809d334b8c9828d6ba556eafefd17fc6f563af5bc165811845216b9872923c2"},"schema_version":"1.0","source":{"id":"1709.04976","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.04976","created_at":"2026-05-18T00:35:08Z"},{"alias_kind":"arxiv_version","alias_value":"1709.04976v1","created_at":"2026-05-18T00:35:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.04976","created_at":"2026-05-18T00:35:08Z"},{"alias_kind":"pith_short_12","alias_value":"7F5OWP4UD2Y7","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_16","alias_value":"7F5OWP4UD2Y7NXQG","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_8","alias_value":"7F5OWP4U","created_at":"2026-05-18T12:31:05Z"}],"graph_snapshots":[{"event_id":"sha256:eec574e9df9daa9e089fafaef8b4b6250544e6beeaa84c862bd1520707d2ee74","target":"graph","created_at":"2026-05-18T00:35:08Z","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":"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","authors_text":"Diego Y\\'a\\~nez, Pedro Mart\\'in","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2017-09-13T10:47:10Z","title":"Geometric clustering in normed planes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.04976","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:ce48c20d99dfb5ef6d611bb9348b3675023c25eaa4dde378ce2a62dceafa016c","target":"record","created_at":"2026-05-18T00:35:08Z","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":"9c18a847782f9063a9d569306cf6995ea899a1e4b43c49a0419440cc0518c880","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2017-09-13T10:47:10Z","title_canon_sha256":"8809d334b8c9828d6ba556eafefd17fc6f563af5bc165811845216b9872923c2"},"schema_version":"1.0","source":{"id":"1709.04976","kind":"arxiv","version":1}},"canonical_sha256":"f97aeb3f941eb1f6de06e4a456b25ee072a4b8b746c40d35f38092c29edb2dbc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f97aeb3f941eb1f6de06e4a456b25ee072a4b8b746c40d35f38092c29edb2dbc","first_computed_at":"2026-05-18T00:35:08.689164Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:35:08.689164Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"midg+C8+496c4LwF28dWhAiKASE123oAfqWrF3PTBE7lIMtvn7ADxqavLdCxlBjNdIII0cpyPy6NKf/t3NVMCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:35:08.689593Z","signed_message":"canonical_sha256_bytes"},"source_id":"1709.04976","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ce48c20d99dfb5ef6d611bb9348b3675023c25eaa4dde378ce2a62dceafa016c","sha256:eec574e9df9daa9e089fafaef8b4b6250544e6beeaa84c862bd1520707d2ee74"],"state_sha256":"f6d16909d49fd48c4215fa123e0c870c18e41e51c313d92fae29c9a2804a687f"}