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Let ξ 1 , ξ 2 , ... be a sequence of independent copies of a random vector in R d having an absolutely continuous distribution. Consider a random walk S i : = ξ 1 + ⋯ + ξ i , and let C n , d : = conv (0 , S 1 , S 2 , ... , S n) be the convex hull of the first n + 1 points it has visited. The polytope C n , d is called k-neighborly if for any indices 0 ≤ i 1 < ⋯ < i k ≤ n the convex hull of the k points S i 1 , ... , S i k is a (k - 1) -dimensional face of C n , d . We study the probability that C n , d is k-neighborly in various high-dimensional asymptotic regimes, i.e. when n, d, and possibly also k diverge to ∞ . There is an explicit formula for the expected number of (k - 1) -dimensional faces of C n , d which involves Stirling numbers of both kinds. Motivated by this formula, we introduce a distribution, called the Lah distribution, and study its properties. In particular, we provide a combinatorial interpretation of the Lah distribution in terms of random compositions and records, and explicitly compute its factorial moments. Limit theorems which we prove for the Lah distribution imply neighborliness properties of C n , d . This yields a new class of random polytopes exhibiting phase transitions parallel to those discovered by Vershik and Sporyshev, Donoho and Tanner for random projections of regular simplices and crosspolytopes. [ABSTRACT FROM AUTHOR]

We consider the simplices K n A = { x ∈ R n + 1 : x 1 ≥ x 2 ≥ ⋯ ≥ x n + 1 , x 1 - x n + 1 ≤ 1 , x 1 + ⋯ + x n + 1 = 0 } and K n B = { x ∈ R n : 1 ≥ x 1 ≥ x 2 ≥ ⋯ ≥ x n ≥ 0 } , which are called the Schläfli orthoschemes of types A and B, respectively. We describe the tangent cones at their j-faces and compute explicitly the sums of the conic intrinsic volumes of these tangent cones at all j-faces of K n A and K n B . This setting contains sums of external and internal angles of K n A and K n B as special cases. The sums are evaluated in terms of Stirling numbers of both kinds. We generalize these results to finite products of Schläfli orthoschemes of type A and B and, as a probabilistic consequence, derive formulas for the expected number of j-faces of the Minkowski sums of the convex hulls of a finite number of Gaussian random walks and random bridges. Furthermore, we evaluate the analogous angle sums for the tangent cones of Weyl chambers of types A and B and finite products thereof. [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 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]

Let X1, . . ., Xn be a standard normal sample in Rd. We compute exactly the expected volume of the Gaussian polytope conv [X1, . . ., Xn], the symmetric Gaussian polytope conv [±X1, . . .,±Xn], and the Gaussian zonotope [0,X1] + · · · + [0,Xn] by exploiting their connection to the regular simplex, the regular cross-polytope, and the cube with the aid of Tsirelson’s formula. The expected volumes of these random polytopes are given by essentially the same expressions as the intrinsic volumes and external angles of the regular polytopes. For all of these quantities, we obtain asymptotic formulae which are more precise than the results which were known before. More generally, we determine the expected volumes of some heteroscedastic random polytopes including conv [l1X1, . . ., lnXn] and conv[±l1X1, . . .,±lnXn], where l1, . . ., ln ≥ 0 are parameters, and the intrinsic volumes of the corresponding deterministic polytopes. Finally, we relate the kth intrinsic volume of the regular simplex Δn−1 to the expected maximum of independent standard Gaussian random variables ξ1, . . ., ξn given that the maximum has multiplicity k. Namely, we show that Vk(Δn−1) = (2π)k/2/k! · lim ε↓0 ε1−kE[max{ξ1, . . ., ξn}1{ξ(n) −ξ(n−k+1) ≤ε}], where ξ(1) ≤ ·· · ≤ ξ(n) denote the order statistics. A similar result holds for the cross-polytope if we replace ξ1, . . ., ξn with their absolute values. [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 n points $$X_1,\ldots ,X_n$$ in $$\mathbb {R}^d$$ and denote their convex hull by $${\Pi }$$ . We prove a number of inclusion-exclusion identities for the system of convex hulls $${\Pi }_I:=\mathrm{conv}(X_i: i\in I)$$ , where I ranges over all subsets of $$\{1,\ldots ,n\}$$ . For instance, denoting by $$c_k(X)$$ the number of k-element subcollections of $$(X_1,\ldots ,X_n)$$ whose convex hull contains a point $$X\in \mathbb {R}^d$$ , we prove that for all X in the relative interior of $${\Pi }$$ . This confirms a conjecture of Cowan (Adv Appl Probab 39(3):630-644, 2007) who proved the above formula for almost all X. We establish similar results for the number of polytopes $${\Pi }_J$$ containing a given polytope $${\Pi }_I$$ as an r-dimensional face, thus proving another conjecture of Cowan (Discrete Comput Geom 43(2):209-220, 2010). As a consequence, we derive inclusion-exclusion identities for the intrinsic volumes and the face numbers of the polytopes $${\Pi }_I$$ . The main tool in our proofs is a formula for the alternating sum of the face numbers of a convex polytope intersected by an affine subspace. This formula generalizes the classical Euler-Schläfli-Poincaré relation and is of independent interest. [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]

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]