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This claim is true and tight for k=0 (this is Rado's centerpoint theorem), as well as for k = d-1 (trivial). Bukh et al. showed the existence of a (d-2)-flat at depth (d-1) n / (2d-1) - O(1) (the case k = d-2).\n  In this paper we concentrate on the case k=1 (the case of \"centerlines\"), in which "},"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":"1107.3421","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2011-07-18T12:54:49Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"ea9dd4b43230d2b7c25f388a378d5c89c3f0846a19becddf2ef7f3b94b621e26","abstract_canon_sha256":"bd1fe5eddc4526d9e7a4d53554329da13ce218fbed1b4f5ec4137846a5801911"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:56:35.370366Z","signature_b64":"ZQ3gftG4k3ONTPGJ6Jun2B0QJpPdHXLZWGwFyPyKhFPVcf4whZgePPXWdvuHJvGeWqmTGVSM6kCARla6krSuCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"688290a65d7471e9b4a39d25e6f2e99a5d2447971f0f6fa19addb4106b866c22","last_reissued_at":"2026-05-18T03:56:35.369946Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:56:35.369946Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Upper bounds for centerlines","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.CG","authors_text":"Boris Bukh, Gabriel Nivasch","submitted_at":"2011-07-18T12:54:49Z","abstract_excerpt":"In 2008, Bukh, Matousek, and Nivasch conjectured that for every n-point set S in R^d and every k, 0 <= k <= d-1, there exists a k-flat f in R^d (a \"centerflat\") that lies at \"depth\" (k+1) n / (k+d+1) - O(1) in S, in the sense that every halfspace that contains f contains at least that many points of S. 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Bukh et al. showed the existence of a (d-2)-flat at depth (d-1) n / (2d-1) - O(1) (the case k = d-2).\n  In this paper we concentrate on the case k=1 (the case of \"centerlines\"), in which "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.3421","kind":"arxiv","version":3},"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":"1107.3421","created_at":"2026-05-18T03:56:35.370003+00:00"},{"alias_kind":"arxiv_version","alias_value":"1107.3421v3","created_at":"2026-05-18T03:56:35.370003+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.3421","created_at":"2026-05-18T03:56:35.370003+00:00"},{"alias_kind":"pith_short_12","alias_value":"NCBJBJS5ORY6","created_at":"2026-05-18T12:26:37.096874+00:00"},{"alias_kind":"pith_short_16","alias_value":"NCBJBJS5ORY6TNFD","created_at":"2026-05-18T12:26:37.096874+00:00"},{"alias_kind":"pith_short_8","alias_value":"NCBJBJS5","created_at":"2026-05-18T12:26:37.096874+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/NCBJBJS5ORY6TNFDTUS6N4XJTJ","json":"https://pith.science/pith/NCBJBJS5ORY6TNFDTUS6N4XJTJ.json","graph_json":"https://pith.science/api/pith-number/NCBJBJS5ORY6TNFDTUS6N4XJTJ/graph.json","events_json":"https://pith.science/api/pith-number/NCBJBJS5ORY6TNFDTUS6N4XJTJ/events.json","paper":"https://pith.science/paper/NCBJBJS5"},"agent_actions":{"view_html":"https://pith.science/pith/NCBJBJS5ORY6TNFDTUS6N4XJTJ","download_json":"https://pith.science/pith/NCBJBJS5ORY6TNFDTUS6N4XJTJ.json","view_paper":"https://pith.science/paper/NCBJBJS5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1107.3421&json=true","fetch_graph":"https://pith.science/api/pith-number/NCBJBJS5ORY6TNFDTUS6N4XJTJ/graph.json","fetch_events":"https://pith.science/api/pith-number/NCBJBJS5ORY6TNFDTUS6N4XJTJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NCBJBJS5ORY6TNFDTUS6N4XJTJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NCBJBJS5ORY6TNFDTUS6N4XJTJ/action/storage_attestation","attest_author":"https://pith.science/pith/NCBJBJS5ORY6TNFDTUS6N4XJTJ/action/author_attestation","sign_citation":"https://pith.science/pith/NCBJBJS5ORY6TNFDTUS6N4XJTJ/action/citation_signature","submit_replication":"https://pith.science/pith/NCBJBJS5ORY6TNFDTUS6N4XJTJ/action/replication_record"}},"created_at":"2026-05-18T03:56:35.370003+00:00","updated_at":"2026-05-18T03:56:35.370003+00:00"}