  Let $K \in \R^d$ be a convex body, and assume that $L$ is a randomly rotated
and shifted integer lattice. Let $K_L$ be the convex hull of the (random)
points $K \cap L$. The mean width $W(K_L)$ of $K_L$ is investigated. The
asymptotic order of the mean width difference $W(\l K)-W((\l K)_L)$ is
maximized by the order obtained by polytopes and minimized by the order for
smooth convex sets as $\l \to \infty$.
