Let X 1 , . . . , X N , N > n, be independent random points in R n , distributed according to the so-called beta or beta-prime distribution, respectively. We establish threshold phenomena for the volume, intrinsic volumes, or more general measures of the convex hulls of these random point sets, as the space dimension n tends to infinity. The dual setting of polytopes generated by random halfspaces is also investigated.

Let X 1 , . . . , X n be independent random points that are distributed according to a probability measure on R d and let P n be the random convex hull generated by X 1 , . . . , X n (n ≥ d + 1). Natural classes of probability distributions are characterized for which, by means of Blaschke-Petkantschin formulae from integral geometry, one can show that the mean facet number of P n is strictly monotonically increasing in n.

The beta polytope P n, is the convex hull of n i.i.d. random points distributed in the unit ball of R according to a density proportional to (1 − ||x|| 2 ) if > −1 (in particular, = 0 corresponds to the uniform distribution in the ball), or uniformly on the unit sphere if = −1. We show that the expected normalized volumes of high-dimensional beta polytopes exhibit a phase transition and we describe its shape. We derive analogous results for the intrinsic volumes of beta polytopes and, when = 0, their number of vertices.

Let Z be the typical cell of a stationary Poisson hyperplane tessellation in R d . The distribution of the number of facets f (Z) of the typical cell is investigated. It is shown, that under a well-spread condition on the directional distribution, the quantity n 2 d−1 n P(f (Z) = n) is bounded from above and from below. When f (Z) is large, the isoperimetric ratio of Z is bounded away from zero with high probability.These results rely on one hand on the Complementary Theorem which provides a precise decomposition of the distribution of Z and on the other hand on several geometric estimates related to the approximation of polytopes by polytopes with fewer facets.From the asymptotics of the distribution of f (Z), tail estimates for the so-called Φ content of Z are derived as well as results on the conditional distribution of Z when its Φ content is large.

We investigate the maximal degree in a Poisson-Delaunay graph in R d , d ≥ 2, over all nodes in the window Wρ := ρ 1/d [0, 1] d as ρ goes to infinity. The exact order of this maximum is provided in any dimension. In the particular setting d = 2, we show that this quantity is concentrated on two consecutive integers with high probability. An extension of this result is discussed when d ≥ 3.We summarize here the notation used throughout the text. General notationWe denote by N = {1, 2, . . .} and R + = [0, ∞) the sets of positive integers and nonnegative numbers, respectively. The d-dimensional Euclidean space R d is endowed with the Euclidean norm · and with its d-dimensional Lebesgue measure V d (·). We denote by B d + the set of Borel sets B ⊂ R d such that 0 < V d (B) < ∞. The unit sphere with dimension d − 1 is denoted by S d−1 .

Until now, little was known about properties of small cells in a Poisson hyperplane tessellation. The few existing results were either heuristic or applying only to the two dimensional case and for very specific size measurements and directional distributions. This paper fills this gap by providing a systematic study of small cells in a Poisson hyperplane tessellation of arbitrary dimension, arbitrary directional distribution ϕ and with respect to an arbitrary size measurement Σ. More precisely, we investigate the distribution of the typical cell Z, conditioned on the event {Σ(Z) < a}, where a → 0 and Σ is a size measurement, i.e. a functional on the set of convex bodies which is continuous, not identically zero, homogeneous of degree k > 0, and increasing with respect to set inclusion. We focus on the number of facets and the shape of such small cells. We show in various general settings that small cells tend to minimize the number of facets and that they have a non degenerated limit shape distribution which depends on the size Σ and the directional distribution. We also exhibit a class of directional distribution for which cells with small inradius do not tend to minimize the number of facets.

This paper presents bounds for the best approximation, with respect to the Hausdorff metric, of a convex body K by a circumscribed polytope P with a given number of facets. These bounds are of particular interest if K is elongated. To measure the elongation of the convex set, its isoperimetric ratio V j (K) 1/j V i (K) −1/i is used.

Facets of the convex hull of 𝑛 independent random vectors chosen uniformly at random from the unit sphere in ℝ 𝑑 are studied. A particular focus is given on the height of the facets as well as the expected number of facets as the dimension increases. Regimes for 𝑛 and 𝑑 with different asymptotic behavior of these quantities are identified and asymptotic formulas in each case are established. Extensions of several known results in fixed dimension to the case where dimension tends to infinity are described.