{"paper":{"title":"One-sided epsilon-approximants","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"Boris Bukh, Gabriel Nivasch","submitted_at":"2016-03-17T22:44:50Z","abstract_excerpt":"Given a finite point set $P\\subset\\mathbb{R}^d$, we call a multiset $A$ a one-sided weak $\\varepsilon$-approximant for $P$ (with respect to convex sets), if $|P\\cap C|/|P|-|A\\cap C|/|A|\\leq\\varepsilon$ for every convex set $C$.\n  We show that, in contrast with the usual (two-sided) weak $\\varepsilon$-approximants, for every set $P\\subset \\mathbb{R}^d$ there exists a one-sided weak $\\varepsilon$-approximant of size bounded by a function of $\\varepsilon$ and $d$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.05717","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"}