{"paper":{"title":"Balanced Islands in Two Colored Point Sets in the Plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Birgit Vogtenhuber, David Flores-Pe\\~naloza, Jorge Urrutia, Jose M. D{\\i}az-B\\'a\\~nez, Nieves Atienza, Oswin Aichholzer, Pablo Perez-Lantero, Ruy Fabila-Monroy","submitted_at":"2015-10-07T04:14:36Z","abstract_excerpt":"Let $S$ be a set of $n$ points in general position in the plane, $r$ of which are red and $b$ of which are blue. In this paper we prove that there exist: for every $\\alpha \\in \\left [ 0,\\frac{1}{2} \\right ]$, a convex set containing exactly $\\lceil \\alpha r\\rceil$ red points and exactly $\\lceil \\alpha b \\rceil$ blue points of $S$; a convex set containing exactly $\\left \\lceil \\frac{r+1}{2}\\right \\rceil$ red points and exactly $\\left \\lceil \\frac{b+1}{2}\\right \\rceil$ blue points of $S$. Furthermore, we present polynomial time algorithms to find these convex sets. In the first case we provide a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.01819","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"}