  Let $K$ be a convex body in $\mathbb{R}^n$ and $f : \partial K \rightarrow
\mathbb{R}_+$ a continuous, strictly positive function with
$\int\limits_{\partial K} f(x) d \mu_{\partial K}(x) = 1$. We give an upper
bound for the approximation of $K$ in the symmetric difference metric by an
arbitrarily positioned polytope $P_f$ in $\mathbb{R}^n$ having a fixed number
of vertices. This generalizes a result by Ludwig, Sch\"utt and Werner $[36]$.
The polytope $P_f$ is obtained by a random construction via a probability
measure with density $f$. In our result, the dependence on the number of
vertices is optimal. With the optimal density $f$, the dependence on $K$ in our
result is also optimal.
