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Consider a finite collection of affine hyperplanes in R d . The hyperplanes dissect R d into finitely many polyhedral chambers. For a point x ∈ R d and a chamber P the metric projection of x onto P is the unique point y ∈ P minimizing the Euclidean distance to x. The metric projection is contained in the relative interior of a uniquely defined face of P whose dimension is denoted by dim (x , P) . We prove that for every given k ∈ { 0 , ... , d } , the number of chambers P for which dim (x , P) = k does not depend on the choice of x, with an exception of some Lebesgue null set. Moreover, this number is equal to the absolute value of the k-th coefficient of the characteristic polynomial of the hyperplane arrangement. In a special case of reflection arrangements, this proves a conjecture of Drton and Klivans [A geometric interpretation of the characteristic polynomial of reflection arrangements. Proc. Amer. Math. Soc. 138(8), 2873–2887 (2010)]. [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]

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]

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]