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

Let Pn be an n-dimensional regular polytope from one of the three infinite series (regular simplices, regular crosspolytopes, and cubes). Project Pn onto a random, uniformly distributed linear subspace of dimension d = 2. We prove that the expected number of k-dimensional faces of the resulting random polytope is an increasing function of n. As a corollary, we show that the expected number of k-faces of the Gaussian polytope is an increasing function of the number of points used to generate the polytope. Similar results are obtained for the symmetric Gaussian polytope and the Gaussian zonotope. [ABSTRACT FROM AUTHOR]