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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]

We study three families of polyhedral cones whose sections are regular simplices, cubes, and crosspolytopes. We compute solid angles and conic intrinsic volumes of these cones. We show that several quantities appearing in stochastic geometry can be expressed through these conic intrinsic volumes. A list of such quantities includes internal and external solid angles of regular simplices and crosspolytopes, the probability that a (symmetric) Gaussian random polytope or the Gaussian zonotope contains a given point, the expected number of faces of the intersection of a regular polytope with a random linear subspace passing through its centre, and the expected number of faces of the projection of a regular polytope onto a random linear subspace. [ABSTRACT FROM AUTHOR]

Consider a random simplex [ X 1 , ... , X n ] defined as the convex hull of independent identically distributed (i.i.d.) random points X 1 , ... , X n in R n - 1 with the following beta density: Let J n , k (β) be the expected internal angle of the simplex [ X 1 , ... , X n ] at its face [ X 1 , ... , X k ] . Define J ~ n , k (β) analogously for i.i.d. random points distributed according to the beta ′ density f ~ n - 1 , β (x) ∝ (1 + ‖ x ‖ 2 ) - β , x ∈ R n - 1 , β > (n - 1) / 2. We derive formulae for J n , k (β) and J ~ n , k (β) which make it possible to compute these quantities symbolically, in finitely many steps, for any integer or half-integer value of β . For J n , 1 (± 1 / 2) we even provide explicit formulae in terms of products of Gamma functions. We give applications of these results to two seemingly unrelated problems of stochastic geometry: (i) We compute explicitly the expected f-vectors of the typical Poisson–Voronoi cells in dimensions up to 10. (ii) Consider the random polytope K n , d : = [ U 1 , ... , U n ] where U 1 , ... , U n are i.i.d. random points sampled uniformly inside some d-dimensional convex body K with smooth boundary and unit volume. Reitzner (Adv. Math. 191(1), 178–208 (2005)) proved the existence of the limit of the normalised expected f-vector of K n , d : lim n → ∞ n - (d - 1) / (d + 1) E f (K n , d) = c d · Ω (K) , where Ω (K) is the affine surface area of K, and c d is an unknown vector not depending on K. We compute c d explicitly in dimensions up to d = 10 and also solve the analogous problem for random polytopes with vertices distributed uniformly on the sphere. [ABSTRACT FROM AUTHOR]