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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]

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]

Let U 1 , U 2 , ... be random points sampled uniformly and independently from the d-dimensional upper half-sphere. We show that, as n → ∞ , the f-vector of the (d + 1) -dimensional convex cone C n generated by U 1 , ... , U n weakly converges to a certain limiting random vector, without any normalization. We also show convergence of all moments of the f-vector of C n and identify the limiting constants for the expectations. We prove that the expected Grassmann angles of C n can be expressed through the expected f-vector. This yields convergence of expected Grassmann angles and conic intrinsic volumes and answers thereby a question of Bárány et al. (Random Struct Algorithms 50(1):3–22, 2017. 10.1002/rsa.20644). Our approach is based on the observation that the random cone C n weakly converges, after a suitable rescaling, to a random cone whose intersection with the tangent hyperplane of the half-sphere at its north pole is the convex hull of the Poisson point process with power-law intensity function proportional to ‖ x ‖ - (d + γ) , where γ = 1 . We compute the expected number of facets, the expected intrinsic volumes and the expected T-functional of this random convex hull for arbitrary γ > 0 . [ABSTRACT FROM AUTHOR]

Let X1,⋯,Xn be i.i.d. random points in the d‐dimensional Euclidean space sampled according to one of the following probability densities: fd,β(x)=const·1−∥x∥2β,∥x∥<1,(thebetacase)and f∼d,β(x)=const·1+∥x∥2−β,x∈Rd,(thebeta"case).We compute exactly the expected intrinsic volumes and the expected number of facets of the convex hull of X1,⋯,Xn. Asymptotic formulae were obtained previously by Affentranger [The convex hull of random points with spherically symmetric distributions, 1991]. By studying the limits of the beta case when β↓−1, respectively β↑+∞, we can also cover the models in which X1,⋯,Xn are uniformly distributed on the unit sphere or normally distributed, respectively. We obtain similar results for the random polytopes defined as the convex hulls of ±X1,⋯,±Xn and 0,X1,⋯,Xn. One of the main tools used in the proofs is the Blaschke–Petkantschin formula. [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]

Consider a sequence of partial sums S i = ξ 1 + … + ξ i , 1 ≤ i ≤ n , starting at S 0 = 0 , whose increments ξ 1 , … , ξ n are random vectors in R d , d ≤ n . We are interested in the properties of the convex hull C n : = Conv ( S 0 , S 1 , … , S n ) . Assuming that the tuple ( ξ 1 , … , ξ n ) is exchangeable and a certain general position condition holds, we prove that the expected number of k -dimensional faces of C n is given by the formula E [ f k ( C n ) ] = 2 ⋅ k ! n ! ∑ l = 0 ∞ [ n + 1 d − 2 l ] { d − 2 l k + 1 } , for all 0 ≤ k ≤ d − 1 , where [ n m ] and { n m } are Stirling numbers of the first and second kind, respectively. Further, we compute explicitly the probability that for given indices 0 ≤ i 1 < … < i k + 1 ≤ n , the points S i 1 , … , S i k + 1 form a k -dimensional face of Conv ( S 0 , S 1 , … , S n ) . This is done in two different settings: for random walks with symmetrically exchangeable increments and for random bridges with exchangeable increments. These results generalize the classical one-dimensional discrete arcsine law for the position of the maximum due to E. Sparre Andersen. All our formulae are distribution-free, that is do not depend on the distribution of the increments ξ k 's. The main ingredient in the proof is the computation of the probability that the origin is absorbed by a joint convex hull of several random walks and bridges whose increments are invariant with respect to the action of direct product of finitely many reflection groups of types A n − 1 and B n . This probability, in turn, is related to the number of Weyl chambers of a product-type reflection group that are intersected by a linear subspace in general position. [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]

We study random polytopes of the form [ X 1 , ... , X n ] defined as convex hulls of independent and identically distributed random points X 1 , ... , X n in R d with one of the following densities: f d , β (x) = c d , β (1 − ‖ x ‖ 2) β , ‖ x ‖ < 1 , (beta distribution, β > − 1) or f ˜ d , β (x) = c ˜ d , β (1 + ‖ x ‖ 2) − β , x ∈ R d , (beta' distribution, β > d / 2). Here, c d , β and c ˜ d , β are normalizing constants. This setting also includes the uniform distribution on the unit sphere and the standard normal distribution as limiting cases. We derive exact and asymptotic formulae for the expected number of k -faces of [ X 1 , ... , X n ] for arbitrary k ∈ { 0 , 1 , ... , d − 1 }. We prove that for any such k this expected number is strictly monotonically increasing with n. Also, we compute the expected internal and external angles of these polytopes at faces of every dimension and, more generally, the expected conic intrinsic volumes of their tangent cones. By passing to the large n limit in the beta' case, we compute the expected f -vector of the convex hull of Poisson point processes with power-law intensity function. Using convex duality, we derive exact formulae for the expected number of k -faces of the zero cell for a class of isotropic Poisson hyperplane tessellations in R d. This family includes the zero cell of a classical stationary and isotropic Poisson hyperplane tessellation and the typical cell of a stationary Poisson–Voronoi tessellation as special cases. In addition, we prove precise limit theorems for this f -vector in the high-dimensional regime, as d → ∞. Finally, we relate the d -dimensional beta and beta' distributions to the generalized Pareto distributions known in extreme-value theory. [ABSTRACT FROM AUTHOR]