Variance Asymptotics and Scaling Limits for Random Polytopes

Let K be a convex set in R d and let K $lambda$ be the convex hull of a homogeneous Poisson point process P $lambda$ of intensity $lambda$ on K. When K is a simple polytope, we establish scaling limits as $lambda$ $rightarrow$ $infty$ for the boundary of K $lambda$ in a vicinity of a vertex of K and we give variance asymptotics for the volume and k-face functional of K $lambda$, k $in$ {0, 1, ..., d -- 1}, resolving an open question posed in [18]. The scaling limit of the boundary of K $lambda$ and the variance asymptotics are described in terms of a germ-grain model consisting of cone-like grains pinned to the extreme points of a Poisson point process on R d--1 $times$ R having intensity $sqrt$ de dh dhdv.