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arXiv · 1512.02449

Random approximation and the vertex index of convex bodies

Silouanos Brazitikos·Giorgos Chasapis·Labrini Hioni
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Abstract

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 symmetric case.

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Silouanos Brazitikos, Giorgos Chasapis, Labrini Hioni. 2015-12-08. Random approximation and the vertex index of convex bodies. https://arxiv.org/abs/1512.02449

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