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 .
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