Poisson Polytopes

We prove the central limit theorem for the volume and the f-vector of the Poisson random polytope η in a fixed convex polytope P ⊂ R d. Here, η is the convex hull of the intersection of a Poisson process X of intensity η with P. 1. Introduction and main results. Let K ⊂ R d be a convex set of volume 1. Assume that X = X(η) is a Poisson point process in R d of intensity η. The intersection of K with X(η) consists of uniformly distributed random points X 1 ,. .. , X N (where N is a random variable). Define the Poisson polytope η , as the convex hull [X 1 ,. .. , X N ] = [K ∩ X(η)]. The study of properties of random convex hulls is a classical subject in stochas-tic geometry and dates back to 1864. Due to the geometric nature of the available methods, for over one hundred years, investigations mainly concentrated on the expectation of functionals of random convex hulls such as volume or number of vertices; see, for example, the survey of Weil and Wieacker [24]. The first distributional results were only proven twenty years ago. In 1988, Groeneboom [14] obtained the central limit theorem (CLT) for the number of ver-tices of the Poisson polytope when the convex body K is the planar disc. In 1994, a CLT for the area of a random polygon in the planar disc was proven by Hs-ing [16]. Recently, this was generalized to arbitrary dimensions by Reitzner [19], who established a CLT for V (η), the volume of the Poisson polytope, and for f ((η), the number of-dimensional faces of the Poisson polytope, when the body K ⊂ R d has smooth boundary. The situation seems to be much more involved when the underlying convex set is a polytope P. In the planar case, when P is a convex polygon, a CLT for the number of vertices f 0 ((η) was proven by Groeneboom [14] and a CLT for the area of η by Cabo and Groeneboom [12], but it seems that the stated variances are incorrect (see the discussion in Buchta [11]). The main result of the present paper is the central limit theorem for the Poisson polytope η for all dimensions d ≥ 2, when the mother body is a polytope in R d .