Recursive Scheme for Angles of Random Simplices, and Applications to Random Polytopes

Consider a random simplex [X1,…,Xn]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[X_1,\ldots ,X_n]$$\end{document} defined as the convex hull of independent identically distributed (i.i.d.) random points X1,…,Xn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_1,\ldots ,X_n$$\end{document} in Rn-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{n-1}$$\end{document} with the following beta density: Let Jn,k(β)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_{n,k}(\beta )$$\end{document} be the expected internal angle of the simplex [X1,…,Xn]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[X_1,\ldots ,X_n]$$\end{document} at its face [X1,…,Xk]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[X_1,\ldots ,X_k]$$\end{document}. Define J~n,k(β)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{J}}_{n,k}(\beta )$$\end{document} analogously for i.i.d. random points distributed according to the beta′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$'$$\end{document} density f~n-1,β(x)∝(1+‖x‖2)-β,x∈Rn-1,β>(n-1)/2.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{f}}_{n-1,\beta } (x) \propto (1+\Vert x\Vert ^2)^{-\beta }, x\in \mathbb {R}^{n-1}, \beta > ({n-1})/{2}.$$\end{document} We derive formulae for Jn,k(β)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_{n,k}(\beta )$$\end{document} and J~n,k(β)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{J}}_{n,k}(\beta )$$\end{document} which make it possible to compute these quantities symbolically, in finitely many steps, for any integer or half-integer value of β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}. For Jn,1(±1/2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_{n,1}(\pm 1/2)$$\end{document} 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 Kn,d:=[U1,…,Un]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{n,d} := [U_1,\ldots ,U_n]$$\end{document} where U1,…,Un\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_1,\ldots ,U_n$$\end{document} 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 Kn,d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{n,d}$$\end{document}: limn→∞n-(d-1)/(d+1)Ef(Kn,d)=cd·Ω(K),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \lim _{n\rightarrow \infty } n^{-{({d-1})/({d+1})}}{\mathbb {E}}{\mathbf {f}}(K_{n,d}) = {\mathbf {c}}_d \cdot \Omega (K),$$\end{document} where Ω(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega (K)$$\end{document} is the affine surface area of K, and cd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {c}}_d$$\end{document} is an unknown vector not depending on K. We compute cd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {c}}_d$$\end{document} explicitly in dimensions up to d=10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=10$$\end{document} and also solve the analogous problem for random polytopes with vertices distributed uniformly on the sphere.


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 R n−1 with probability distribution μ. Consider a random simplex defined as their convex hull: [X 1 , . . . , X n ] := {λ 1 X 1 + · · · + λ n X n : λ 1 + · · · + λ n = 1, λ 1 ≥ 0, . . . , λ n ≥ 0}.
For the class of distributions studied here, this simplex is non-degenerate (i.e., has a non-empty interior) a.s. Let β ([X 1 , . . . , X k ], [X 1 , . . . , X n ]) denote the internal angle of the simplex [X 1 , . . . , X n ] at its (k − 1)-dimensional face [X 1 , . . . , X k ]. Similarly, we denote by γ ([X 1 , . . . , X k ], [X 1 , . . . , X n ]) the external (or normal) angle of [X 1 , . . . , X n ] at [X 1 , . . . , X k ]. The exact definitions of internal and external angles will be recalled in Sect. 4.1; see also the book [35] for an extensive account of stochastic geometry. We agree to choose the units of measurement for angles in such a way that the full-space angle equals 1. We shall be interested in the expected values of the above-defined angles. The special case when μ is a multivariate normal distribution has been studied in [12,13,21], where the following theorem has been demonstrated.

Beta and Beta Distributions
In the present paper we shall be interested in the case when μ belongs to one of the following two remarkable families of probability distributions introduced by Miles [27] and studied by Ruben and Miles [33]. A random vector in R d has a d-dimensional beta distribution with parameter β > −1 if its Lebesgue density is Here, x = (x 2 1 + · · · + x 2 d ) 1/2 denotes the Euclidean norm of the vector x = (x 1 , . . . , x d ) ∈ R d . Similarly, a random vector in R d has beta distribution with parameter β > d/2 if its Lebesgue density is given bỹ The following particular cases are of special interest: (a) The beta distribution with β = 0 is the uniform distribution in the unit ball B d := {x ∈ R d : x ≤ 1}. (b) The weak limit of the beta distribution as β ↓ −1 is the uniform distribution on the unit sphere S d−1 := {x ∈ R d : x = 1}; see [19]. In the following, we write f d,−1 for the uniform distribution on S d−1 , and the results of the present paper apply to the case β = −1. (c) The standard normal distribution on R d is the weak limit of both beta and beta distributions (after suitable rescaling) as β → +∞; see [20,Lem. 1.1]. (d) The beta distributionf n−1,n/2 on R n−1 with β = n/2 is the image of the uniform distribution on the upper half-sphere S n−1 + under the so-called gnomonic projection [18,Prop. 2.2]; see also [17] for further applications of this observation. The triangular arrays J n,k (β) andJ n,k (β) appeared in [20] together with the closely related arrays I n,k (α) andĨ n,k (α) that are essentially the expected external angles of beta and beta simplices; see Theorems 1.2 and 1.3, below. It has been shown in [20] that many quantities appearing in stochastic geometry can be expressed in terms of I n,k (β),Ĩ n,k (β) and J n,k (β),J n,k (β). An incomplete list of such quantities is as follows: (a) The expected f -vectors of beta and beta polytopes. The beta polytopes are defined as random polytopes of the form P expected f -vectors of the random polytopes in the half-sphere; see [17] for a detailed study of these models. (d) Expected f -vector of the typical Poisson-Voronoi cell. (e) Constants appearing in the work of Reitzner [30] on the asymptotics of the expected f -vectors of random polytopes approximating smooth convex bodies. (f) External and internal angles of the regular simplex with n vertices at its k-vertex faces. These coincide with the corresponding expected angles of the random Gaussian simplex [13,21], and are given by I n,k (+∞) := lim β↑+∞ I n,k (β) and J n,k (+∞) := lim β↑+∞ J n,k (β), respectively.
While there exist explicit formulae for I n,k (α) andĨ n,k (α) (see Sect. 1.4), no general formulae are known for J n,k (β) andJ n,k (β) except in some special cases. For example, we have J 3,1 (β) = 1/2 because the sum of angles in any plane triangle equals half the full angle. For general n ∈ N, it always holds that J n,n (β) = 1 and J n,n−1 (β) = n/2, and all these formulae are valid in the beta case, too. A general combinatorial formula forJ n,k (n/2) was derived in [17], where it was used to compute the expected f -vector of the Poisson zero polytope. For n = 4 and n = 5, explicit formulae for J n,k (β) were derived in [16] by a method not allowing for an extension to higher dimensions. The main results of the present paper can be summarised as follows. In Sect. 2, we derive a formula which enables us to compute J n,k (β) andJ n,k (β) symbolically for half-integer β, and numerically for all admissible β. The main work for this formula has been done in [17,20], while the main contribution of the present paper is its explicit statement and demonstration of some consequences. The latter will be done in Sect. 3, where we apply the formula to compute (among other examples) the expected fvectors of typical Poisson-Voronoi cells and the constants that appeared in the work of Reitzner [30] on random polytopes approximating convex bodies, in dimensions up to 10.

Expected External Angles
The following two theorems define the quantities I n,k (α) andĨ n,k (α) and relate them to the expected external angles of beta and beta simplices. They are special cases of Theorems 1.6 and 1.16 in [20], respectively.
where for α > 0 we definẽ Usually, it will be more convenient to work with angle sums rather than with individual angles, which is why we introduce the quantities Note that I n,n (α) =Ĩ n,n (α) = 1.

Algorithm for Computing Expected Internal-Angle Sums
In the next proposition we state relations which enable us to express the quantities J n,k (β) through the quantities I n,k (α). The proof will be given in Sect. 4.2, where we shall also discuss similarity between these relations and McMullen's angle-sum relations [23,24] for deterministic polytopes.
The following equation, which follows from (6) and (7) by taking the arithmetic mean, is more efficient for computational purposes since it contains less terms than (10): For example, the first few non-trivial values of the internal-angles vector Generalising these formulae, we can prove the following Theorem 2.3 For every β ≥ −1, n ∈ N, and k ∈ {1, . . . , n} we have that 2J n,k (β) − δ n,k is equal to where δ n,k is Kronecker's delta, and the sum is taken over all integer tuples (n 0 , n 1 , . . . , n ) such that n = n 0 > n 1 > · · · > n ≥ k and such that n − n i is even for all i ∈ {1, . . . , }.
The quantitiesJ n,k (β) can be computed in a similar manner. We putJ 1,1 (β) = 1 and then use the recursive formulã which follows from (8). Alternatively, one can use the more efficient formulã which follows from (8) and (9) by taking their arithmetic mean. The next two theorems are similar to Theorems 2.2 and 2.3. We omit their straightforward proofs.

Relations in Matrix Form
Let us write the relation (7) in the form where δ nk denotes Kronecker's delta. Introducing the new variable γ := β +(n −1)/2 that ranges in the interval [(n − 3)/2, +∞), we can write This relation has the advantage that now the J-term does not contain n, which allows to state it in matrix form. Take some N ∈ N, γ ≥ (N − 3)/2, and introduce the N × N matrices A and B with the following entries; Note that both A and B are lower-triangular matrices with 1's on the diagonal. Then, (12) states that AB = E, where E is the N × N identity matrix. Since this implies that BA = E, we arrive at the following relation which is dual to (12):

Arithmetic Properties of Expected Internal-Angle Sums
At the moment, we do not have a general formula for J n,k (β) andJ n,k (β) which is "nicer" than what is given in Theorems 2. (a) If α is odd, then I n,k (α) is rational.
Using the above theorem together with the results of Sect. 2.1, we shall prove the following result on the J n,k (β)'s.
(a) If 2β + n is even, then J n,k (β) is a rational number.
Symbolic computations we performed with the help of Mathematica 11 strongly suggest that in the case when n − k is odd, part (b) can be strengthened as follows:

Conjecture 2.9
If both 2β + n and n − k are odd, then J n,k (β) is a number of the form qπ −(n−k−1) with some rational q.
Conjecture 2.9 states that J n,k (β) has sometimes much simpler form than the one suggested by Theorems 2.2 and 2.3. For example, when computing J 7,2 (−1), we can use the formula which follows from (11). The involved values are given by so that, a priori, we expect J 7,2 (−1) to be a linear combination of 1, π −2 , π −4 over Q. A posteriori, it turns out that J 7,2 (−1) = 113537407/(16128000π 4 ) is a rational multiple of π −4 , while the remaining terms cancel. We were not able to explain this strange cancellation using Theorems 2.2 and 2.3. It is therefore natural to conjecture that there is a "nicer" formula for J n,k (β) than the ones given in these theorems. The results for the quantitiesĨ n,k (α) andJ n,k (β) are analogous. We state them without proofs.
where the r i 's are rational numbers.
(a) If 2β − n is odd, thenJ n,k (β) is a rational number.
In the case when k is even, our symbolic computations suggest the following stronger version of (b):

Conjecture 2.12
If both 2β − n and k are even, thenJ n,k (β) is a number of the form

Special Cases and Applications
In this section we present several special cases of the above results and their applications to some problems of stochastic geometry. The symbolic computations were performed using Mathematica 11. For the vector of the expected internal angles we use the notation . . , J n,n (β)).

Internal Angles of Random Simplices: Uniform Distribution on the Sphere
Let X 1 , . . . , X n be i.i.d. random points sampled uniformly from the unit sphere S n−2 ⊂ R n−1 . Recall that the expected sum of internal angles of the simplex [X 1 , . . . , X n ] at its k-vertex faces is denoted by J n,k (−1). Clearly, The first two non-trivial cases, n = 3 and n = 4 (corresponding to simplices in dimensions 3 and 4), were treated in [16]: The method used there did not allow for an extension to higher dimensions. Using Mathematica 11 and the algorithm described in Sect. 2.1 we recovered these results and, moreover, obtained the following

Internal Angles of Random Simplices: Uniform Distribution in the Ball
Let X 1 , . . . , X n be i.i.d. random points sampled uniformly from the unit ball B n−1 .
The expected sum of internal angles of the simplex [X 1 , . . . , X n ] at its k-vertex faces is J n,k (0). The values of J n,k (0) for n = 1, 2, 3 are the same as in (16). For simplices with n = 4 and n = 5 vertices (corresponding to dimensions d = 3 and 4), the following results were obtained in [16] by a method not extending to higher dimensions: Using Mathematica 11 and the above algorithm we recovered these results and, moreover, obtained the following

Typical Poisson-Voronoi Cells
Let P 1 , P 2 , . . . be the points of a Poisson point process on R d with constant intensity 1.
The typical Poisson-Voronoi cell is a random polytope which, for our purposes, can be defined as follows: The typical Poisson-Voronoi cell is one of the classical objects of stochastic geometry; see [8,9,15,28,29,35] for reviews and the works of Meijering [25], Gilbert [11], and Miles [26] for important early contributions. We shall be interested in the expected f -vector of V d denoted by To the best of our knowledge, explicit formulae for the complete vector Ef(V d ) have been known only in dimensions d = 2 and 3: Taking k = 0, we recover (18). Formula (19), together with the algorithm for computation ofJ n,k (β), allows us to compute E f k (V d ) in finitely many steps. Using Mathematica 11, we have done this in dimensions d ∈ {2, . . . , 10}. As a result, we recovered (17) and, moreover, obtained the following

Theorem 3.3 The expected f -vector of the typical Poisson-Voronoi cell is given by
where the coefficients q i are rational.
Proof of (a) Let d be even. Recall that (x) is integer if x > 0 is integer, and is a rational multiple of √ π if x > 0 is half-integer. It follows from (20) thatĨ ∞,m (d) is rational. Also, by Theorem 2.11 (a),  1 (mod 2). The claim follows.
In fact, a closer look at the values collected in Theorem 3.3 suggests the following conjecture which is a consequence of Conjecture 2.12.

Random Polytopes Approximating Smooth Convex Bodies
Let U 1 , U 2 , . . . be independent random points distributed uniformly in the ddimensional convex body K . Denote the convex hull of n such points by K n,d = [U 1 , . . . , U n ]. Asymptotic properties of K n,d , as n → ∞, have been very much studied starting with the work of Rényi and Sulanke [31,32] (see, for example, [15,34]) and we shall not attempt to review the vast literature on this topic. In particular, regarding the f -vector of K n,d , this development culminated in the work of Reitzner who proved the following result [30, p. 181]. If the boundary of K is of differentiability class C 2 and the Gaussian curvature κ(x) > 0 is positive at every boundary point x ∈ ∂ K , then for every k ∈ {0, 1, . . . , d − 1}, where (K ) := ∂ K κ(x) 1/(d+1) dx is the so-called affine surface area of K , and c d,0 , . . . , c d,d−1 are certain strictly positive constants not depending on K . In [30], (21) is stated without the term involving Vol d K , for which it is necessary to assume that K has unit volume. The general case follows from the following scaling property of the affine surface area:  = (c d,0 , . . . , c d,d−1 ); but we have not succeeded in getting an explicit expression". Our aim is to provide explicit expressions for c d for all d ≤ 10. In the following, it will be convenient to take K := B d (which is possible since c d does not depend on K ) and use the notation (22) for all k ∈ {0, 1, . . . , d − 1}. Here, we recall that P 0 n,d = [X 1 , . . . , X n ] is the convex hull of n i.i.d. random points X 1 , . . . , X n distributed uniformly in the ball B d . Note also that the affine surface area of the unit ball coincides with its usual surface area: For d = 2, the value of C 2,0 = C 2,1 has been identified by Rényi and Sulanke [31, Satz 3] who proved that If d ∈ N is arbitrary and k = d − 1, Affentranger [2] (see his Corollary 1 on p. 366, the formula for c 3 on p. 378, and take q = 0) proved the following formula for C d,d−1 : Note also that an exact formula for the number of facets of a convex hull of N i.i.d. points sampled uniformly from the ball B d has been obtained by Buchta and Müller [6] (see their Theorem 3 on page 760), but it requires some work to analyse its asymptotic behaviour as N → ∞. In [20, Rem. 1.9], it has been shown that for all d ∈ N and k ∈ {0, . . . , d − 1}, In the special case k = d − 1, (24) reduces to (23)

Random Polytopes with Vertices on the Sphere
Similarly, one can consider random polytopes approximating a convex body K and having vertices on the boundary of K . Here, we restrict ourselves to the case K = B d , so that we are interested in the random polytope P −1 n,d defined as the convex hull of n points X 1 , . . . , X n chosen uniformly at random on the unit sphere S d−1 , d ≥ 2. In [20,Rem. 1.9], it has been shown that for all k ∈ {0, . . . , d − 1}. In the special case k = d − 1, it was previously shown by Affentranger [2] (see his Corollary 1 on p. 366 and the formula for c 3 on p. 378, this time with q = −1) and Buchta et al. [7] (see their formula forF (d) n on p. 231) that where the second equality follows from the duplication formula for the Gamma function. This formula for C * d,d−1 is a special case of (25) since J d,d (−1/2) = 1. Using (25) together with the algorithm for computing J d,k+1 (−1/2), we obtain the following Observe that the first entry of each vector is C * d,0 = 1 for all d ∈ N. This is trivial because all points X 1 , . . . , X n are vertices of P −1 n,d . Yet, in the above table, the constant 1 appeared as a result of a non-trivial computation of J d,1 (−1/2). On the one hand, this gives evidence for the correctness of the algorithm. On the other hand, it can be used to give an explicit formula for J d,1 (−1/2), as we shall show in the next section.
Proof The argument follows essentially the approach sketched by Hug [15, pp. 209-210]. Consider N i.i.d. points uniformly distributed in the unit ball B d . Denote their convex hull by P 0 N ,d . As N → ∞, the random polytope P 0 N ,d approaches the unit ball. In particular, E Vol d P 0 N ,d converges to κ d , the volume of B d . The speed of convergence has been identified by Wieacker [36]; see also [2] for similar results on general beta polytopes and [1,19] for exact formulae for the expected volume. In particular, it is known that as N → ∞; see, for example Corollary 1 on p. 366 of [2] and the formula for c 5 on p. 378, with q = 0. The left-hand side is closely related to the expected number of vertices of P 0 N ,d via Efron's identity which states that Indeed, the N -th point is a vertex of P 0 N ,d if and only if it is outside the convex hull of the remaining N −1 points. If we condition on the first N −1 points, then the probability that the last point is a vertex is (κ d − Vol d P 0 N −1,d )/κ d . Taking expectations proves Efron's identity. From (27) and (28) we deduce that as N → ∞. On the other hand, we know from (22) and (24) (where we take k = 0) that as N → ∞. Equating the constants on the right-hand sides of (29) and (30), resolving w.r.t. J d, 1 (1/2), and simplifying, we arrive at the second formula stated in Theorem 3.8.
The equivalence of both formulae is easily shown using the identity which is equivalent to the Legendre duplication formula for the Gamma function.
Theorem 3.9 For every n ∈ {2, 3, . . .} we have We shall give two independent proofs. The first one is based on (25) (which, as was explained above, generalises (26) obtained independently in [2] and [7]). The second proof relies, among other ingredients, on a formula due to Kingman [22]. The fact that all these formulae lead to the same result can be viewed as an additional evidence for their correctness.

First proof of Theorem 3.9
Recall that P −1 N ,d is the convex hull of N i.i.d. points having the uniform distribution on S d−1 . By a formula derived in [20], we have On the other hand, in the special case when k = 0 we trivially have f 0 (P −1 N ,d ) = N a.s. since every point is a vertex. Hence, the right-hand side equals 1 if k = 0, which yields Replacing d by n completes the proof of the second formula stated in Theorem 3.9. The equivalence to the first formula follows from Legendre's duplication formula (31).
The second proof of Theorem 3.9 uses the following observation of Feldman and Klain [10]. It can be viewed as a special case of a more general result that has been obtained earlier by Affentranger and Schneider [3]. Then, it follows from Theorem 3.10 and Fubini's formula that Let us compute the probability on the right-hand side. Let I L : L → R d−1 be an isometry with I L (0) = 0. By the projection property of the beta densities (see [19,Lem. 4.4]) the points have the density f d−1,0 . That is, these points are uniformly distributed in the unit ball B d−1 . Clearly, these points are i.i.d. We have where the last equality is Efron's identity obtained by conditioning on Y 1 , . . . , Y d and recalling that Y 0 is uniformly distributed in B d−1 . A formula for the expected volume on the right-hand side is well known from the work of Kingman [22,Thm. 7]: where the second equality can be verified using the duplication formula for the Gamma function. Taking everything together and recalling that d = n − 1 completes the proof.

Notation and Facts from Stochastic Geometry
Let us first introduce the necessary notation, referring to the book by Schneider and Weil [35] for an extensive account of stochastic geometry. A polyhedral cone (or just a cone) C ⊂ R d is an intersection of finitely many closed half-spaces whose boundaries pass through the origin. The solid angle of C is defined as where U is a random vector having the uniform distribution on the unit sphere of the smallest linear subspace containing C. For example, the angle of R d is 1, whereas the angle of any half-space is 1/2. Let P ⊂ R d be a d-dimensional convex polytope. Denote by F k (P) the set of its k-dimensional faces, where k ∈ {0, 1, . . . , d}. The set of all faces of P is denoted by F • (P) = d k=0 F k (P). The tangent cone of P at its face F ∈ F k (P) is defined as where f 0 is any point in the relative interior of F, defined as the interior of F taken with respect to its affine hull. The internal angle of P at its face F ∈ F k (P) is defined by β(F, P) := α(T (F, P)).
The normal or external cone of F is defined as the polar cone of T (F, P), that is N (F, P) = {z ∈ R d : z, y ≤ 0 for all y ∈ T (F, P)}.
The normal of external angle of P at its face F ∈ F k (P) is defined by γ (F, P) := α(N (F, P)).

Remark 4.1
It is possible to obtain another proof of Proposition 2.1 using McMullen's non-linear angle-sum relations [23,24]. These state that for every face F ∈ F • (P) of an arbitrary polytope P, where δ F,P = 1 if F = P, and δ F,P = 0 otherwise. Applied to P = P The so-called canonical decomposition of beta distributions, see [33]  Observe that on the right-hand side we have a quantity different from J m,k (β) since the points X 1 , . . . , X m are in R n−1 and do not form a full-dimensional simplex, so that we cannot directly apply the definition of J m,k (β). Taking the expectation of (36) and using the above facts, we obtain n m=k n − k m − k I n,m (2β + n − 1) J m,k β + n − m 2 = 1.

Proofs: Arithmetic Properties
In this section we prove Theorems 2.7 and 2.8. The proofs of Theorems 2.10 and 2.11, being analogous to the proofs of Theorems 2.7 and 2.8, are omitted.

Proof of Theorems 2.7 and 2.8
Recall from Sect. 2.1 that we can express J n,k (β) through the quantities of the form In Propositions 5.4 and 5.6 we shall establish the arithmetic properties of I n,k (α) for integer α ≥ 0. Taken together, these propositions yield Theorem 2.7.
Proof Just recall the following two facts:  When multiplying out the terms under the integral sign, we obtain a finite Q-linear combination of the terms of the form e isϕ (with odd s) and ie isϕ (with even s). The integral of a term of the former type is a rational number since +π/2 −π/2 e isϕ dϕ = e isπ/2 − e −isπ/2 is ∈ Q, s ∈ {±1, ±3, . . .}.
Finally, the term ie i0ϕ must have coefficient 0 since its integral is purely imaginary and we know a priori that I n,k (α) is real. Hence, I n,k (α) is rational.

Case 2:
Let k ∈ {1, . . . , n} be even. Then, αk is also even and c 1,(αk−1)/2 is a rational multiple of 1/π by Lemma 5.1. We can write When multiplying out the terms under the sign of the integral, we obtain a finite Qlinear combination of terms of the form π −1 e isϕ (with even s) and iπ −1 e isϕ (with odd s). The integral of the term π −1 e i0ϕ is 1. By the same analysis as in Case 1, the integrals of all terms with s = 0 are purely imaginary and hence must cancel since we know a priori that I n,k (α) is real. Hence, I n,k (α) is rational.
1, π −2 , π −4 , . . . , π −( p+b) . If both p and b are odd, then by Lemma 5.5 (b) the term is a Q-linear combination of 1, π −2 , π −4 , . . . , π −( p+b) . If the parities of p and b differ, then the term is purely imaginary and can be ignored since we a priori know that I n,k (α) is real, which implies that all such terms must cancel.