Recursive Scheme for Angles of Random Simplices, and Applications to Random Polytopes
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Recursive Scheme for Angles of Random Simplices, and Applications to Random Polytopes Zakhar Kabluchko1 Received: 19 August 2019 / Revised: 12 July 2020 / Accepted: 21 October 2020 © The Author(s) 2020
Abstract 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 Rn−1 with the following beta density: f n−1,β (x) ∝ (1−x2 )β 1{x −1. Let Jn,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+x2 )−β , x ∈ Rn−1 , β > (n − 1)/2. We derive formulae for Jn,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 Jn,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 := [U1 , . . . , Un ] where U1 , . . . , Un 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 : limn→∞ n −(d−1)/(d+1) Ef(K n,d ) = cd · (K ), where (K ) is the affine surface area of K , and cd is an unknown vector not depending on K . We compute cd explicitly in dimensions up to d = 10 and also solve the analogous problem for random polytopes with vertices distributed uniformly on the sphere. Keywords Random polytope · Random simplex · Solid angle · Sum of angles · Beta distribution · Typical Poisson–Voronoi cell Mathematics Subject Classification 52A22 · 60D05 · 52A55 · 52B11 · 60G55 · 52A27
Editor in Charge: Kenneth Clarkson Zakhar Kabluchko [email protected] 1
Institut für Mathematische Stochastik, Westfälische Wilhelms-Universität Münster, Orléans-Ring 10, 48149 Münster, Germany
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Discrete & Computational Geometry
1 Statement of the Problem 1.1 Introduction It is well known that the sum of angles in any plane triangle is constant, whereas the sum of solid d-dimensional angles at the vertices of a d-dimensional simplex is not, starting with dimension d = 3. It is therefore natural to ask what is the “average” angle-sum of a d-dimensional simplex. To define the notion of average, we put a probability measure on the set simplices as follows. Let X 1 , . . . , X n be independent, identically distributed (i.i.d.) random points in Rn−1 with probability distribution μ. Consider a random simplex defined as their convex hull: [X 1 , . . . , X n ] := {λ1 X 1 + · · · + λn X n : λ1 + · · · + λn = 1, λ1 ≥ 0, . . . , λn ≥ 0}. For the class of distributions studied here, this simplex is non-degenerate (i.e
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