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
\({{\rm card}(X)=\lceil\alphan\rceil }\) 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.