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Various mixing properties of β -, β ′ - and Gaussian-Delaunay tessellations in ℝ d − 1 are studied. It is shown that these tessellation models are absolutely regular, or β -mixing. In the β - and the Gaussian case exponential bounds for the absolute regularity coefficients are found. In the β ′ -case these coefficients show a polynomial decay only. In the background are new and strong concentration bounds on the radius of stabilization of the underlying construction. Using a general device for absolutely regular stationary random tessellations, central limit theorems for a number of geometric parameters of β - and Gaussian-Delaunay tessellations are established. This includes the number of k -dimensional faces and the k -volume of the k -skeleton for k ∈ { 0 , 1 , ... , d − 1 }. [ABSTRACT FROM AUTHOR]

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]

The β-Delaunay tessellation in ℝd-1 is a generalization of the classical Poisson-Delaunay tessellation. As a first result of this paper we show that the shape of a weighted typical cell of a βDelaunay tessellation, conditioned on having large volume, is close to the shape of a regular simplex inℝd-1. This generalizes earlier results of Hug and Schneider about the typical (non-weighted) Poisson-Delaunay simplex. Second, the asymptotic behaviour of the volume of weighted typical cells in high-dimensional β-Delaunay tessellation is analysed, as d → ∞. In particular, various high dimensional limit theorems, such as quantitative central limit theorems as well as moderate and large deviation principles, are derived. [ABSTRACT FROM AUTHOR]

The typical cell of a Voronoi tessellation generated by n + 1 uniformly distributed random points on the d-dimensional unit sphere S d is studied. Its f-vector is identified in distribution with the f-vector of a beta' polytope generated by n random points in R d . Explicit formulas for the expected f-vector are provided for any d and the low-dimensional cases d ∈ { 2 , 3 , 4 } are studied separately. This implies an explicit formula for the total number of k-dimensional faces in the spherical Voronoi tessellation as well. [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]

We consider d-dimensional random vectors Y1, . . . ,Yn that satisfy a mild general position assumption a.s. The hyperplanes (Yi−Yj)⊥ (1 ≤ i < j ≤ n) generate a conical tessellation of the Euclidean d-space which is closely related to the Weyl chambers of type An−1. We determine the number of cones in this tessellation and show that it is a.s. constant. For a random cone chosen uniformly at random from this random tessellation, we compute expectations of several geometric functionals. These include the face numbers, as well as the conic intrinsic volumes and the conical quermassintegrals. Under the additional assumption of exchangeability on Y1, . . . ,Yn, the same is done for the dual random cones which have the same distribution as the positive hull of Y1−Y2, . . . , Yn−1−Yn conditioned on the event that this positive hull is not equal to Rd. All these expectations turn out to be distribution-free. Similarly, we consider the conical tessellation induced by the hyperplanes (Yi + Yj)⊥ (1 ≤ i < j ≤ n), (Yi − Yj)⊥ (1 ≤ i < j ≤ n), Yi⊥ (1 ≤ i ≤ n). This tessellation is closely related to the Weyl chambers of type Bn. We compute the number of cones in this tessellation and the expectations of various geometric functionals for random cones drawn from this random tessellation. The main ingredient in the proofs is a connection between the number of faces of the tessellation and the number of faces of the Weyl chambers of the corresponding type that are intersected non-trivially by a certain linear subspace in general position. [ABSTRACT FROM AUTHOR]

In this paper, we prove three fundamental types of limit theorems for the q -norm of random vectors chosen at random in an ℓ p n -ball in high dimensions. We obtain a central limit theorem, a moderate deviations as well as a large deviations principle when the underlying distribution of the random vectors belongs to a general class introduced by Barthe, Guédon, Mendelson, and Naor. It includes the normalized volume and the cone probability measure as well as projections of these measures as special cases. Two new applications to random and non-random projections of ℓ p n -balls to lower-dimensional subspaces are discussed as well. The text is a continuation of [Z. Kabluchko, J. Prochno and C. Thäle, High-dimensional limit theorems for random vectors in ℓ p n -balls, Commun. Contemp. Math.21(1) (2019) 1750092]. [ABSTRACT FROM AUTHOR]

Motivated by problems of hyperbolic stochastic geometry we introduce and study the class of beta-star polytopes. A beta-star polytope is defined as the convex hull of an inhomogeneous Poisson processes on the complement of the unit ball in R d with density proportional to (‖ x ‖ 2 − 1) − β , where ‖ x ‖ > 1 and β > d / 2. Explicit formulas for various geometric and combinatorial functionals associated with beta-star polytopes are provided, including the expected number of k -dimensional faces, the expected external angle sums and the expected intrinsic volumes. Beta-star polytopes are relevant in the context of hyperbolic stochastic geometry, since they are tightly connected to the typical cell of a Poisson-Voronoi tessellation as well as the zero cell of a Poisson hyperplane tessellation in hyperbolic space. The general results for beta-star polytopes are used to provide explicit formulas for the expected f -vector of the typical hyperbolic Poisson-Voronoi cell and the hyperbolic Poisson zero cell. Their asymptotics for large intensities and their monotonicity behaviour is discussed as well. Finally, stochastic geometry in the de Sitter half-space is studied as the hyperbolic analogue to recent investigations about random cones generated by random points on half-spheres in spherical or conical stochastic geometry. [ABSTRACT FROM AUTHOR]

Pick d + 1 points uniformly at random on the unit sphere in R d. What is the expected value of the angle sum of the simplex spanned by these points? Choose n points uniformly at random in the d -dimensional ball. What is the expected number of faces of their convex hull? We answer these and some other, seemingly unrelated, questions of stochastic geometry. To this end, we compute expected internal angles of random simplices whose vertices are independent random points sampled from one of the following d -dimensional distributions: (i) the beta distribution with the density proportional to (1 − ‖ x ‖ 2) β , where x is belongs to the unit ball in R d ; (ii) the beta' distribution with the density proportional to (1 + ‖ x ‖ 2) − β on the whole of R d. These results imply explicit formulae for the expected face numbers of the following random polytopes: (a) the typical Poisson-Voronoi cell; (b) the zero cell of the Poisson hyperplane tessellation; (c) beta and beta' polytopes defined as convex hulls of i.i.d. samples from the corresponding distributions. [ABSTRACT FROM AUTHOR]