Variance asymptotics and scaling limits for random polytopes

Let K be a convex set in Rd and let Kλ be the convex hull of a homogeneous Poisson point process Pλ of intensity λ on K. When K is a simple polytope, we establish scaling limits as λ→∞ for the boundary of Kλ in a vicinity of a vertex of K and we give variance asymptotics for the volume and k-face functional of Kλ, k∈{0,1,...,d−1}, resolving an open question posed in [17]. The scaling limit of the boundary of Kλ 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 Rd−1×R having intensity dedhdhdv.