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We give a simple alternative proof of the existence of a $(1/r)$-cutting of the first $n/r$ levels of $\\Arr(H)$, which consists of $O(r)$ semi-unbounded vertical trian"},"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":"1601.04755","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2016-01-18T23:15:22Z","cross_cats_sorted":[],"title_canon_sha256":"99cf1b70c4a2d73b3b150afded8c4f85f55819170663f2358b968dc8ade11017","abstract_canon_sha256":"c4356fcb307b2b355113fda125f343eb32171f1defdad6c38000ee6126e6da53"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:14:44.193851Z","signature_b64":"uEnEYdj8EwpPYZ9/oYMNHIHLTGKuBlGW4x69QCRN++SrbPdNHsUMKYV2ssk2I1FXoL7Kbe1G4c+a2+nS6+OTAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3f34a89609effc347c928fe7b32174c10526fde6dcaaf30476035096ebeef91e","last_reissued_at":"2026-05-18T01:14:44.193209Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:14:44.193209Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Approximating the $k$-Level in Three-Dimensional Plane Arrangements","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Haim Kaplan, Micha Sharir, Sariel Har-Peled","submitted_at":"2016-01-18T23:15:22Z","abstract_excerpt":"$\\renewcommand{\\Re}{{\\rm I\\!\\hspace{-0.025em} R}} \\newcommand{\\SetX}{\\mathsf{X}} \\newcommand{\\eps}{\\varepsilon} \\newcommand{\\VorX}[1]{\\mathcal{V} \\pth{#1}} \\newcommand{\\Polygon}{\\mathsf{P}} \\newcommand{\\IntRange}[1]{[ #1 ]} \\newcommand{\\Space}{\\ovebarline{\\mathsf{m}}} \\newcommand{\\pth}[2][\\!]{#1\\left({#2}\\right)} \\newcommand{\\Arr}{{\\cal A}}$\n  Let $H$ be a set of $n$ planes in three dimensions, and let $r \\leq n$ be a parameter. 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