{"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. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.04755","kind":"arxiv","version":2},"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"}