Expected mean width of the randomized integer convex hull

Binh Hong Ngoc; Matthias Reitzner

paper

Expected mean width of the randomized integer convex hull

arXiv:2003.06864

Abstract

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 $W (ł K) - W ((ł K)_{L})$ is maximized by the order obtained by polytopes and minimized by the order for smooth convex sets as $ł \to \infty$ .