{"paper":{"title":"On the Expected Complexity of Random Convex Hulls","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Sariel Har-Peled","submitted_at":"2011-11-22T21:17:34Z","abstract_excerpt":"In this paper we present several results on the expected complexity of a convex hull of $n$ points chosen uniformly and independently from a convex shape.\n  (i) We show that the expected number of vertices of the convex hull of $n$ points, chosen uniformly and independently from a disk is $O(n^{1/3})$, and $O(k \\log{n})$ for the case a convex polygon with $k$ sides. Those results are well known (see \\cite{rs-udkhv-63,r-slcdn-70,ps-cgi-85}), but we believe that the elementary proof given here are simpler and more intuitive.\n  (ii) Let $\\D$ be a set of directions in the plane, we define a genera"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.5340","kind":"arxiv","version":1},"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"}