  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.
