{"paper":{"title":"Random approximation and the vertex index of convex bodies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Giorgos Chasapis, Labrini Hioni, Silouanos Brazitikos","submitted_at":"2015-12-08T13:10:57Z","abstract_excerpt":"We prove that there exists an absolute constant $\\alpha >1$ with the following property: if $K$ is a convex body in ${\\mathbb R}^n$ whose center of mass is at the origin, then a random subset $X\\subset K$ of cardinality ${\\rm card}(X)=\\lceil\\alpha n\\rceil $ satisfies with probability greater than $1-e^{-n}$ {K\\subseteq c_1n\\,{\\mathrm conv}(X),} where $c_1>0$ is an absolute constant. As an application we show that the vertex index of any convex body $K$ in ${\\mathbb R}^n$ is bounded by $c_2n^2$, where $c_2>0$ is an absolute constant, thus extending an estimate of Bezdek and Litvak for the symme"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.02449","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"}