Face numbers of high-dimensional Poisson zero cells

Abstract

Let Z d \mathcal Z_d be the zero cell of a d d -dimensional, isotropic and stationary Poisson hyperplane tessellation. We study the asymptotic behavior of the expected number of k k -dimensional faces of Z d \mathcal Z_d , as d → ∞ d\to \infty . For example, we show that the expected number of hyperfaces of Z d \mathcal Z_d is asymptotically equivalent to 2 π / 3 d 3 / 2 \sqrt {2\pi /3}\, d^{3/2} , as d → ∞ d\to \infty . We also prove that the expected solid angle of a random cone spanned by d d random vectors that are independent and uniformly distributed on the unit upper half-sphere in R d \mathbb R^{d} is asymptotic to 3 π − d \sqrt 3 \pi ^{-d} , as d → ∞ d\to \infty .