Variance Asymptotics and Scaling Limits for Random Polytopes
Abstract
Let K be a convex set in R d and let K λ be the convex hull of a homogeneous Poisson point process P λ of intensity λ on K. When K is a simple polytope, we establish scaling limits as λ → ∞ for the boundary of K λ in a vicinity of a vertex of K and we give variance asymptotics for the volume and k-face functional of K λ, k ∈ 0, 1, ..., d -- 1, resolving an open question posed in [18]. The scaling limit of the boundary of K λ 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 × R having intensity de dh dhdv.
0