Interaction of Poisson hyperplane processes and convex bodies

Given a stationary and isotropic Poisson hyperplane process and a convex body K in ${\mathbb R}^d$ , we consider the random polytope defined by the intersection of all closed half-spaces containing K that are bounded by hyperplanes of the process not intersecting K. We investigate how well the expected mean width of this random polytope approximates the mean width of K if the intensity of the hyperplane process tends to infinity.

A Uniform Asymptotical Upper Bound for the Variance of a Random Polytope in a Simple Polytope

The present paper contains a sketch of the proof of an upper bound for the variance of the number of hyperfaces of a random polytope when the mother body is a simple polytope. Thus we verify a weaker version of the result in [1] stated without a proof. The article is published in the author’s wording.

On the approximation of a ball by random polytopes

Let be a sequence of independent and identically distributed random vectors drawn from the d-dimensional unit ball Bd and let Xn be the random polytope generated as the convex hull of a1,· ··, an. Furthermore, let Δ(Xn): = Vol (BdXn) be the volume of the part of the ball lying outside the random polytope. For uniformly distributed ai and 2 we prove that the limiting distribution of Δ(Xn)/Ε (Δ (Xn)) for n → ∞ (satisfies a 0–1 law. In particular, we show that Var for n → ∞. We provide analogous results for spherically symmetric distributions in Bd with regularly varying tail. In addition, we indicate similar results for the surface area and the number of facets of Xn.

On the approximation of a ball by random polytopes

Letbe a sequence of independent and identically distributed random vectors drawn from thed-dimensional unit ballBdand letXnbe the random polytope generated as the convex hull ofa1,· ··,an.Furthermore, let Δ(Xn): = Vol (BdXn) be the volume of the part of the ball lying outside the random polytope. For uniformly distributedaiand2 we prove that the limiting distribution of Δ(Xn)/Ε(Δ(Xn)) forn→ ∞ (satisfies a 0–1 law. In particular, we show that Varforn→ ∞. We provide analogous results for spherically symmetric distributions inBdwith regularly varying tail. In addition, we indicate similar results for the surface area and the number of facets ofXn.