Random approximation and the vertex index of convex bodies

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 $${1-e^{-c_1n}}$$ $$K\subseteq c_2n\, {\rm conv}(X),$$ where $${c_1, c_2 > 0}$$ are absolute constants. As an application we show that the vertex index of any convex body K in $${{\mathbb R}^n}$$ is bounded by $${c_3n^2}$$ , where $${c_3 > 0}$$ is an absolute constant, thus extending an estimate of Bezdek and Litvak for the symmetric case.