Random approximation and the vertex index of convex bodies

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 $Xsubset K$ of cardinality ${rm card}(X)=lceilalpha nrceil $ satisfies with probability greater than $1-e^{-n}$ {Ksubseteq 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.