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.
Categories: math.MG