The randomized integer convex hull

Let $K$ 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) lattice points in $K$, the random integer convex hull.
In this talk we will present old and new results on the volume and the mean width of $K_L$. The asymptotic order of the volume, resp. mean width difference is extremal for polytopes on the one hand side and for smooth convex sets on the other side.