Angle Sums of Schläfli Orthoschemes

We consider the simplices KnA={x∈Rn+1:x1≥x2≥⋯≥xn+1,x1-xn+1≤1,x1+⋯+xn+1=0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} K_n^A=\{x\in {\mathbb {R}}^{n+1}:x_1\ge x_2\ge \cdots \ge x_{n+1},x_1-x_{n+1}\le 1,\,x_1+\cdots +x_{n+1}=0\} \end{aligned}$$\end{document}and KnB={x∈Rn:1≥x1≥x2≥⋯≥xn≥0},\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} K_n^B=\{x\in {\mathbb {R}}^n:1\ge x_1\ge x_2\ge \cdots \ge x_n\ge 0\}, \end{aligned}$$\end{document}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 KnA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_n^A$$\end{document} and KnB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_n^B$$\end{document}. This setting contains sums of external and internal angles of KnA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_n^A$$\end{document} and KnB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_n^B$$\end{document} 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.

In the present paper, we evaluate certain angle sums or, more generally, sums of conic intrinsic volumes, of the Schläfli orthoschemes. For a polytope P, let F j (P) denote the set of all j-dimensional faces of P. The tangent cone of P at its jdimensional face F is the convex cone T F (P) defined by T F (P) = {x ∈ R n : v + εx ∈ P for some ε > 0}, where v is any point in F not belonging to a face of smaller dimension. We explicitly compute the conic intrinsic volumes of the tangent cones of the Schläfli orthoschemes at their j-dimensional faces and, in particular, the sum of the conic intrinsic volumes over all such faces. The k-th conic intrinsic volume of a convex cone C, denoted by υ k (C), is a spherical or conic analogue of the usual intrinsic volume of a convex set and will be formally introduced in Sect. 2.2. Among other results, we will show that where the numbers n k and n k are the Stirling numbers of the first and second kind, respectively.
Furthermore, we will compute the analogous angle (and conic intrinsic volume) sums for the tangent cones of Weyl chambers of type A and B which are convex cones in R n defined by A (n) := {x ∈ R n : x 1 ≥ x 2 ≥ · · · ≥ x n } and B (n) := {x ∈ R n : x 1 ≥ x 2 ≥ · · · ≥ x n ≥ 0}.
The corresponding formulas are given by where the numbers B [n, k] and B{n, k} denote the B-analogues of the Stirling numbers of the first and second kind, respectively, which we will formally introduce in Sect. 2.
3. An application of (1.1) and (1.2) to a problem of compressed sensing will be given in Sect. 3.6. Observe that in the special cases k = n and k = j, (1.1) and (1.2) yield formulas for the sums of the internal and external angles of Schläfli orthoschemes and Weyl chambers. We will generalize the above results to finite products of Schläfli orthoschemes and finite products of Weyl chambers leading to rather complicated formulas in terms of coefficients in the Taylor expansion of a certain function. The main results on angle and conic intrinsic volume sums will be stated in Sect. 3.2. As a probabilistic interpretation of these results, we consider convex hulls of Gaussian random walks and random bridges in Sect. 3.4. The expected numbers of j-faces of the convex hull of a single Gaussian random walk or a Gaussian bridge in R d (even in a more general non-Gaussian setting) were already evaluated in [19]. Our general result on the angle sums of products of Schläfli orthoschemes yields a formula for the expected number of j-faces of the Minkowski sum of several convex hulls of Gaussian random walks or Gaussian random bridges.
It turns out that the tangent cones of the Schläfli orthoschemes (and of the Weyl chambers) are essentially products of Weyl chambers of type A and B. We will derive (1.1) and (1.2) as a special case of a more general Proposition 3.8 stated in Sect. 3.3. This proposition gives a formula for the sum of the conic intrinsic volumes of a product of Weyl chambers in terms of the generalized Stirling numbers of the first and second kind. The main ingredients in the proof of this proposition are the known formulas for the conic intrinsic volumes of Weyl chambers; see e.g. [20,Thm. 4.2] or [14, Thm. 1.1].

Preliminaries
In this section we collect notation and facts from convex geometry and combinatorics. The reader may skip this section at first reading and return to it when necessary.

Facts from Convex Geometry
For a set M ⊂ R n denote by lin M (respectively, aff M) its linear (respectively, affine) hull, that is, the minimal linear (respectively, affine) subspace containing M. A set C ⊂ R n is called a (convex) cone if λ 1 x 1 + λ 2 x 2 ∈ C for all x 1 , x 2 ∈ C and λ 1 , λ 2 ≥ 0. Thus, pos M is the minimal cone containing M. The dual cone of a cone C ⊂ R n is defined as where · , · denotes the Euclidean scalar product. We will make use of the following simple duality relation that holds for arbitrary x 1 , . . . , x m ∈ R n : (2.1) A polyhedral set P ⊂ R d is an intersection of finitely many closed half-spaces (whose boundaries need not pass through the origin). A bounded polyhedral set is called polytope. A polyhedral cone is an intersection of finitely many closed half-spaces whose boundaries contain the origin and therefore a special case of a polyhedral set. The faces of P (of arbitrary dimension) are obtained by replacing some of the halfspaces, whose intersection defines the polyhedral set, by their boundaries and taking the intersection. For k ∈ {0, 1, . . . , d}, we denote the set of k-dimensional faces of a polyhedral set P by F k (P). Furthermore, we denote the number of k-faces of P by f k (P) := |F k (P)|. The tangent cone of P at a face F ∈ F k (P) is defined by where v is any point in the relative interior of F. It is known that this definition does not depend on the choice of v. The normal cone of P at the face F is defined as the dual of the tangent cone, that is It is easy to check that given a face F of a cone C, the corresponding normal cone where L ⊥ denotes the orthogonal complement of a linear subspace L.

Conic Intrinsic Volumes and Angles of Polyhedral Sets
Now let us introduce some geometric functionals of cones that we are going to consider.
The following facts are mostly taken from [2,Sect. 2]; see also [23,Sect. 6.5]. At first, we define the conical intrinsic volumes which are the analogues of the usual intrinsic volumes in the setting of conic or spherical geometry. Let C ⊂ R n be a polyhedral cone and g be an n-dimensional standard Gaussian random vector. Then, for k ∈ {0, . . . , n}, the k-th conic intrinsic volume of C is defined by Here, C denotes the metric projection on C, that is C (x) is the vector in C minimizing the Euclidean distance to x ∈ R n . The conic intrinsic volumes of a cone C form a probability distribution on {0, 1, . . . , dim C}, that is The Gauss-Bonnet formula [2,Cor. 4.4] states that for every cone C that is not a linear subspace, which implies that k=1,3,5,...
Furthermore, the conic intrinsic volumes satisfy the product rule where C 1 ×· · ·×C m is the Cartesian product of C 1 , . . . , C m . The product rule implies that the generating polynomial of the intrinsic volumes of C, defined by The solid angle (or just angle) of a cone C ⊂ R n is defined as where Z is uniformly distributed on the unit sphere in the linear hull lin C. Equivalently, we can take a random vector Z having a standard Gaussian distribution on the ambient linear subspace lin C. For a d-dimensional cone C ⊂ R n , where d ∈ {1, . . . , n}, the d-th conical intrinsic volume coincides with the solid angle of C, that is The internal angle of a polyhedral set P at a face F is defined as the solid angle of its tangent cone: The external angle of P at a face F is defined as the solid angle of the normal cone of F with respect to P, that is The conic intrinsic volumes of a cone C ⊂ R n can be computed in terms of the internal and external angles of its faces as follows: Let W n−k ⊂ R n be random linear subspace having the uniform distribution on the Grassmann manifold of all (n − k)-dimensional linear subspaces. Then, following Grünbaum [16], the Grassmann angle γ k (C) of a cone C ⊂ R n is defined as for k ∈ {0, . . . , n}. The Grassmann angles do not depend on the dimension of the ambient space, that is, if we embed C in R N where N ≥ n, the Grassmann angle will be the same. If C is not a linear subspace, then γ k (C)/2 is also known as the k-th conic quermassintegral U k (C) of C, see [17, (1)-(4)], or as the half-tail functional h k+1 (C), see [3]. The conic intrinsic volumes and the Grassmann angles are known to satisfy the linear relation see [2, (2.10)], provided C is not a linear subspace.

Stirling Numbers and Their Generating Functions
In this section, we are going to recall the definitions of various kinds of Stirling numbers and their generating functions. As mentioned in the introduction, these numbers appear in various results presented in this paper.
The (signless) Stirling number of the first kind n k is defined as the number of permutations of the set {1, . . . , n} having exactly k cycles. Equivalently, these numbers can be defined as the coefficients of the polynomial for n ∈ N 0 , with the convention that n k = 0 for n ∈ N 0 , k / ∈ {0, . . . , n}, and 0 0 = 1. By [22, (1.9),(1.15)], the Stirling numbers of the first kind can also be represented as the following sum: The B-analogues of the Stirling numbers of the first kind, denoted by B[n, k], are defined as the coefficients of the polynomial for n ∈ N 0 and, by convention, B[n, k] = 0 for k / ∈ {0, . . . , n}. These numbers appear as entry A028338 in [24]. The exponential generating functions of the arrays The Stirling number of the second kind n k is defined as the number of partitions of the set {1, . . . , n} into k non-empty subsets. Similarly to (2.10), the Stirling numbers of the second kind can be represented as the following sum: (2.14) They appear as entry A039755 in [24] and were studied by Suter [25]. The exponential generating functions of the arrays n k n,k≥0 and (B{n, k}) n,k≥0 are given by see entry A039755 in [24] and also [4,5] for combinatorial proofs of both identities, which should be compared to the formulas (2.9) and (2.11) for n k and their B-analogues B [n, k].
More generally, it is possible to define the r -Stirling numbers of the first and second kinds. For r ∈ N 0 , the (signless) r -Stirling number of the first kind, denoted by n k r , is defined as the number of permutations of the set {1, . . . , n} having k cycles such that the numbers 1, 2, . . . , r are in distinct cycles; see [8, (1)]. The r -Stirling number of the second kind, denoted by n k r , is defined as the number of partitions of the set {1, . . . , n} into k non-empty disjoint subsets such that the numbers 1, 2, . . . , r are in distinct subsets; see [8, (2)]. Obviously, for r ∈ {0, 1}, the r -Stirling numbers of the first and second kinds coincide with the classical Stirling numbers, respectively. The r -Stirling numbers were introduced by Carlitz [9,10] under the name weighted Stirling numbers.
The exponential generating functions in one and two variables of the r -Stirling numbers of the first kind are given by  Both can easily be verified by comparing the generating functions. Let us also mention that besides the r -Stirling numbers there is another construction, the generalized twoparameter Stirling numbers [21] (see also [6]) containing Stirling numbers of both types and their B-analogues as special cases corresponding to the parameters (d, a) = (1, 0) and (d, a) = (2, 1), respectively. We will not use this general construction here.

Schläfli Orthoschemes
The polytopes we are interested in this paper are called Schläfli orthoschemes. As mentioned in the introduction, the Schläfli orthoscheme of type B in R n is defined as Note that, for convenience, we set K B 0 := {0}. Similarly, the Schläfli orthoscheme of type A in R n+1 is defined as the convex hull of the (n + 1)-dimensional vectors P 0 , P 1 , . . . , P n+1 , where P 0 = (0, 0, . . . , 0) and It is not difficult to check that Again, we put K A 0 = {0}. The index shift from type B to type A will turn out to be convenient since K A n ⊂ R n+1 is an n-dimensional polytope. In fact, the Schläfli orthoschemes of type A and type B are simplices since they are convex hulls of n + 1 affinely independent vectors. The Schläfli orthoscheme of type B was already considered by Gao and Vitale [13] who among other things evaluated the classical intrinsic volumes of K B n . Similar calculations for the Schläfli orthoscheme of type A were made by Gao [12].
The definition of the Schläfli orthoscheme can be motivated by a connection to random walks and random bridges. In fact, consider Gaussian random matrices G B ∈ R d×n and G A ∈ R d×(n+1) , that is, random matrices having independent and standard Gaussian distributed entries. Then G B K B n has the same distribution as the convex hull of a d-dimensional random walk S 0 := 0, S 1 , . . . , S n with Gaussian increments. Similarly, G A K A n has the same distribution as the convex hull of a d-dimensional Gaussian random bridge S 0 := 0, S 1 , . . . , S n , S n+1 = 0 which is essentially a Gaussian random walk conditioned on the event that it returns to 0 in the (n + 1)-st step. We will explain these facts in Sect. 3.4 in more detail.

Sums of Conic Intrinsic Volumes in Weyl Chambers and Schläfli Orthoschemes
In this section, we state the main results of this paper concerning the sums of the conic intrinsic volumes of the tangent cones of Schläfli orthoschemes of type A and B and their products. The same is done for Weyl chambers of type A and B and their products. Our first result concerning the Schläfli orthoschemes of types A and B is the following theorem.
As a consequence, we can derive formulas for the sums of the internal and external angles of K B n and K A n at their j-faces F.

Corollary 3.2 For j ∈ {0, . . . , n}, the sum of the internal angles is given by
while the sum of the external angles are given by Proof The sums of the internal angles follow from Theorem 3.1 with k = n, since K B n and K A n are both n-dimensional polytopes. In the case of external angles, we use the fact that the maximal linear subspaces contained in both T F (K B n ) and , and similarly for K A n . Using Theorem 3.1 with k = j completes the proof.
We obtain similar results for the tangent cones of Weyl chambers of types A and B, which are the fundamental domains of the reflection groups of the respective type; see, e.g., [18]. More concretely, a Weyl chamber of type B (or B n ) is the polyhedral cone The Weyl chamber of type A (or A n−1 ) is the polyhedral cone Theorems 3.1 and 3.3 are special cases of Proposition 3.8 which we shall state in Sect. 3.3 and which gives a formula for sums of the conic intrinsic volumes of a mixed product of Weyl chambers of both types A and B. We will give an application of Theorems 3.1 and 3.3 to a problem of compressed sensing in Sect. 3.6. It has been pointed to us by an anonymous referee that there is a similarity between Theorem 3.3 and the formulas, derived by Amelunxen and Lotz [2, Sect. 6.1.3], for the sums of k-th conic intrinsic volumes of j-dimensional faces in the fans of reflection arrangements. Despite the seeming similarity between the formulas, no direct connection between the quantities under interest seems to exist, the proofs are different and, in fact, even the right-hand sides of the formulas include Stirling numbers in a different order and cannot be reduced to each other in a simple way.
For k = n and k = j, Theorem 3.3 yields the following corollary on the sums of internal and external angles of B (n) and A (n) .

Corollary 3.4
For j ∈ {0, . . . , n}, the sums of internal angles of B (n) and A (n) are given by while the sums of external angles are given by

Finite Products of Schläfli Orthoschemes and Weyl Chambers
The above theorems can be extended to finite products of Schläfli orthoschemes and Weyl chambers. Let b ∈ N and define K B : . Taylor expansion of a function f : R → R around 0 and The proof of Theorem 3.5 is postponed to Sect. 4.2. For finite products of Weyl chambers W B := B (n 1 ) × · · · × B (n b ) and W A := A (n 1 ) × · · · × A (n b ) , we obtain the following theorems.  j, b, (n 1 , . . . , n b )).
The proof of Theorem 3.6 is similar to that of Theorem 3.5 and will be omitted. In the proof of Theorem 3.5 we will observe that if we additionally sum over all possible n 1 , . . . , n b with fixed sum n, the formulas in terms of Taylor coefficients simplify as follows.
The proof is postponed to Sect. 4.3.

Method of Proof of Theorems 3.1 and 3.3
The main ingredient in proving Theorems 3.1 and 3.3 is the following proposition.
For l = (l 1 , . . . , l j+b ) such that l 1 , . . . , l j ∈ N, l j+1 , . . . , l j+b ∈ N 0 , and l 1 + · · · + l j+b = n we define Then, for all k ∈ {0, . . . , n}, we have We will prove this proposition in Sect. 4.1 by computing the generating function of the intrinsic volumes. An alternative proof of Proposition 3.8 can be found in the arXiv version of this paper, see [15,Sect. 4.2]. In order to see that Theorems 3.1 and 3.3 follow from Proposition 3.8, we describe the collections of tangent cones of the Schläfli orthoschemes and Weyl chambers of types A and B at their corresponding faces.

Schläfli Orthoschemes of Type B
The faces of K B n (and of any polytope in general) are obtained by replacing some of the linear inequalities in its defining conditions by equalities. Thus, each j-face of K B n is determined by a collection J := {i 0 , . . . , i j } of indices 0 ≤ i 0 < i 1 < · · · < i j ≤ n and given by Note that for i 0 = 0, no x i is required to be 1. Similarly, for i j = n, no x i is required to be 0. Take a point x = (x 1 , . . . , x n ) ∈ relint F J . For this point, all inequalities in the defining condition of F J are strict. By definition, the tangent cone of K B n at F J is given by It follows that which is isometric to the product where the polyhedral cones i ∈ N 0 , are the Weyl chambers of type B and A, respectively. We arrive at the following lemma.
If the isometry type of some cone appears with some multiplicity in one collection, then it appears with the same multiplicity in the other collections.

Schläfli Orthoschemes of Type A
Now, we consider the tangent cones of Schläfli orthoschemes of type A. Recall that Note that, unlike in the B-case, the simplex K A n (which has dimension n) is contained in R n+1 . For us it will be easier to consider the following unbounded set: where ⊕ denotes the orthogonal sum. Thus, there is a one-to-one correspondence between the j-faces of K A n and the ( j + 1)-faces of K A n given by F → L n+1 ⊕ F. Furthermore, for every j-face F of K A n we have a relation between the tangent cones of K A n and K A n given by Now, consider the collection of tangent cones T F ( K A n ), where F ∈ F j ( K A n ) for some j ∈ {1, . . . , n + 1} and n ∈ N 0 , more closely. The faces of K A n are obtained by replacing some inequalities in the defining conditions of K A n by equalities. Thus, there are two types of j-faces of K A n for j ∈ {1, . . . , n + 1}. The j-faces of the first type are of the form Note that for j = 1, this reduces to the 1-face {x ∈ R n+1 : x 1 = · · · = x n+1 }. To determine the tangent cone at F 1 , take some point in the relative interior of this face. For this point, all inequalities in the defining condition of F 1 are strict. Call this point x = (x 1 , . . . , x n+1 ) ∈ relint F 1 . By definition, we have It follows that where l 1 , . . . , l j ∈ N satisfy l 1 +· · ·+l j = n + 1 and are given by The j-faces of K A n of the second type are of the form The defining condition consists of j +1 groups of equalities and the additional condition x 1 − x n+1 = 1. Again, take a point x ∈ relint F 2 . For this point all inequalities in the defining condition of F 2 are strict. Hence, the tangent cone is given by where in the last step we merged two groups of inequalities. Hence, T F 2 ( K A n ) is isometric to A (l 1 +l j+1 ) × A (l 2 ) × · · · × A (l j ) , where l 1 , . . . , l j+1 ∈ N are such that l 1 + · · · + l j+1 = n + 1, that is they form a composition of n + 1 into j + 1 parts.
where each cone of the above collection is repeated l 1 times (or taken with multiplicity l 1 ).
Then, Theorem 3.1 can be deduced from Proposition 3.8 and Lemma 3.9 (in the B-case), respectively Lemma 3.10 (in the A-case), as follows.
Proof of Theorem 3.1 assuming Proposition 3. 8 We start with the B-case. For j ∈ {0, . . . , n} and k ∈ {0, . . . , n}, we have where we used Lemma 3.9 in the first step and Proposition 3.8 with b = 2 in the second step.
The A-case requires slightly more work. Using the identity υ k+1 (K A n ⊕ L n+1 ) = υ k (K A n ) and (3.1), we obtain Applying Lemma 3.10 j + 1 times with multiplicity l 1 replaced by l 1 , . . . , l j+1 , we can observe that the collection of tangent cones T F ( K A n ), where F runs through all ( j + 1)-faces of F j+1 ( K A n ) and each cone of this collection is repeated j + 1 times, coincides (up to isometry) with the collection of cones A (l 1 ) × · · · × A (l j+1 ) , l 1 , . . . , l j+1 ∈ N, l 1 + · · · + l j+1 = n + 1, where each cone is taken with multiplicity l 1 + · · · + l j+1 = n + 1. Therefore, we arrive at where we used Proposition 3.8. Dividing both sides by j + 1 yields the claim.
Note that for i j = n, no x i 's are required to be 0, and for j = 0, we obtain the 0-dimensional face {0}. In order to determine the tangent cone T F J (B (n) ) take a point x = (x 1 , . . . , x n ) ∈ relint F J . Again, this point satisfies the defining conditions of F J with inequalities replaced by strict inequalities. Thus, the tangent cone is given by The above reasoning yields the following lemma.

Weyl Chambers of Type A
For n ∈ N recall that For j ∈ {1, . . . , n} every j-face of A (n) is determined by a collection J := {i 1 , . . . , i j−1 } of indices 1 ≤ i 1 < · · · < i j−1 ≤ n − 1, and given by Note that for j = 1, we obtain the 1-face {x ∈ R n : x 1 = · · · = x n }. In order to determine the tangent cone T F J (A (n) ) consider a point x = (x 1 , . . . , x n ) ∈ relint F J . In a fashion similar to the case of a B-type Weyl chamber, we can characterize the tangent cone of A (n) at F J as follows: This yields the following analogue of Lemma 3.11.

Lemma 3.12 The collection of tangent cones T F (A (n) ), where F runs through the set of all j-faces F j (A (n) ), coincides with the collection of polyhedral cones
A (l 1 ) × · · · × A (l j ) , l 1 + · · · + l j = n, l 1 , . . . , l j ∈ N.
Proof of Theorem 3.3 assuming Proposition 3. 8 We start with the B-case. For j ∈ {0, . . . , n} and k ∈ {0, . . . , n}, we have where we used Lemma 3.11 in the first step, Proposition 3.8 with b = 1 in the second step, and the formulas in (2.23) in the last step.
In the A-case, for j ∈ {1, . . . , n} and k ∈ {1, . . . , n}, we have where we used Lemma 3.12 in the first step and Proposition 3.8 with b = 0 in the second step. For j = 0 or k = 0 the formula is evidently true as well.

Identities for the Generalized Stirling Numbers
Let us also mention that Proposition 3.8, combined with (2.2) and (2.3), yields the following identities for the generalized Stirling numbers. The full proof can be found in the arxiv version of this paper, see [15].

Corollary 3.13
For n ∈ N, j ∈ {0, . . . , n}, and 2b ∈ N 0 , the following identities hold: Note that (3.5) is a special case of the orthogonality relation between the r -Stirling numbers proven by Broder [8,Thm. 25]. Relation (3.4) is also known for b = 0, 1, the numbers on the right-hand side being the Lah numbers.

Expected Face Numbers: Convex Hulls of Gaussian Random Walks and Bridges
The above theorems on the angle sums of the tangent cones of Schläfli orthoschemes yield results on the expected number of faces of Gaussian random walks and random bridges, and their Minkowski sums. Consider independent d-dimensional standard Gaussian random vectors 1 , X and let n 1 + · · · + n b = n ≥ d. For every i ∈ {1, . . . , b} we define a random walk (S Consider the convex hulls of these random walks: The following theorem gives a formula for the expected number of j-faces of the Minkowski sum of b such convex hulls defined by Theorem 3.14 Let 0 ≤ j < d ≤ n be given and define C where we recall that j, b, (n 1 , . . . , n b )) The same formula holds for the Minkowski sum of Gaussian random bridges which are essentially Gaussian random walks under the condition that they return to 0 in the last step. To state it, consider independent d-dimensional standard Gaussian random vectors 2 , . . . , X 1 , X 2 , . . . , X and define the random walks S Define the convex hulls of the Gaussian bridges by Theorem 3.15 Let 0 ≤ j < d ≤ n be given and C (1) n b be as above. Then, we have j, b, (n 1 , . . . , n b )).
For a single convex hull (b = 1), the expected number of j-faces of C (1) n and C (1) n is already known (in a more general case), see [19, Thms. 1.2 and 5.1], and given by This formula is a special case of Theorems 3.14 and 3.15 with b = 1 and n 1 replaced by n.

Method of Proof of Theorems 3.14 and 3.15
The main ingredient in the proof of the named theorems is the following lemma which is due to Affentranger and Schneider [1, (5)].

Lemma 3.16
Let P ⊂ R n be a convex polytope with non-empty interior and G ∈ R d×n be a Gaussian random matrix, that is, its entries are independent and standard normal random variables. Then, for all 0 ≤ j < d ≤ n we have where the Grassmann angles γ d were defined in (2.7).
In fact, Affentranger and Schneider [1] proved this formula for a random orthogonal projection of P, while the fact that Gaussian matrices yield the same result follows from a result of Baryshnikov and Vitale [7]. Due to the relation between Grassmann angles and conic intrinsic volumes stated in (2.8), the lemma can be written as Using (2.4), it also follows that under the conditions of Lemma 3. 16 we have . . , n}, are independent standard Gaussian random variables. Then, we claim that G B K B has the same distribution as the Minkowski sum C (1) Similarly, for a Gaussian matrix G A ∈ R d×(n+b) , we claim that G A K A has the same distribution as C (1) n 1 + · · · + C (b) n b . In order to see this, consider the case of a single Schläfli orthoscheme K B n 1 first. Let G 1 be a Gaussian matrix. We know that the Schläfli orthoscheme K B n 1 is the simplex given as the convex hull of the n 1 -dimensional vectors  1, 1, 0, . . . , 0), . . . , (1, 1, 1 . . . , 1).
Similarly, we consider a Gaussian matrix G (1) A ∈ R d×(n 1 +1) for the Schläfli orthoscheme K A n 1 . We know that K A n 1 is the convex hull of the (n 1 + 1)-dimensional vectors P 0 = (0, 0, . . . , 0) and 1, 1, . . . , 1), Thus, in the same way we get The product case follows from the following observation. Let G B ∈ R d×n be a Gaussian matrix and n 1 + · · · + n b = n. Then, we can represent G B as the row of b independent matrices G B = G We can represent each vector x ∈ R n as the column of b vectors, i.e., x = (x (1) , . . . , x (b) ) , where x (i) ∈ R n i for i = 1, . . . , b. Then, we easily observe that It follows that In the same way we obtain for a Gaussian matrix G A ∈ R d×(n+b) , n 1 + · · · + n b = n, that Now, we can finally prove Theorems 3.14 and 3.15.
Proof of Theorem 3.14 Let 1 ≤ j < d ≤ n and G B ∈ R d×n be a Gaussian matrix. As we already observed, 2l + 1, j, b, (n 1 , . . . , n b )), where we used Theorem 3.5 in the last step.

Proof of Theorem 3.15
Let 1 ≤ j < d ≤ n and let G A ∈ R d×(n+b) be a Gaussian matrix. We have already seen that G A K A d = C (1) Although the polytope K A ⊂ R n+b is only n-dimensional, we can still apply Lemma 3.16, and therefore also (3.6), to the ambient linear subspace lin K A since the Grassmann angles do not depend on the dimension of the ambient linear subspace. Combining this with Theorem 3.5, we obtain 2l + 1, j, b, (n 1 , . . . , n b )), which completes the proof.

Application to Compressed Sensing
Let us briefly mention an application of Theorems 3.1 and 3.3 to compressed sensing. Donoho and Tanner [11] have considered the following problem. Let x = (x 1 , . . . , x n ) be an unknown signal belonging to some polyhedral set P ⊂ R n and let G : R n → R k be a Gaussian matrix, where k ≤ n. We would like to recover the signal x from its image y = Gx. Following Donoho and Tanner [11], we denote the event that such a recovery is uniquely possible by Unique(G, x, P) : The system y = Gx has a unique solution x = x in P.
The recovery is uniquely possible if and only if (x + ker G) ∩ P = {x}. Assume that x ∈ relint F for some j-face F ∈ F j (P) of P. Then, the following equivalence holds: Using this observation, Donoho and Tanner [11] computed the probability of unique recovery explicitly in the cases when P = R n + is the non-negative orthant or P = [0, 1] n is the unit cube. By the way, the equivalence (3.7) was stated by Donoho and Tanner [11, Lems. 2.1 and 5.1] in these special cases, but it easily generalizes to any polyhedral set. We are now going to use Theorems 3.1 and 3.3 to compute the probabilities of unique signal recovery in the case when P is a Weyl chamber or a Schläfli orthoscheme, which corresponds to natural isotonic constraints frequently imposed in statistics.

Proposition 3.17
Let 0 ≤ j ≤ k ≤ n and let x ∈ R n be a random signal constructed as above (where we put x = 0 if j = 0). If G : R n → R k is a Gaussian random matrix, then it holds that Proof Using the above equivalence (3.7) and the construction of x, we obtain that P Unique(G, x, B (n) ) = n j −1 Since ker G is rotationally invariant, and thus a uniformly distributed (n − k)dimensional linear subspace, we conclude that the probability on the right-hand side coincides with 1 − γ k (T F (B (n) )), where γ k (T F (B (n) )) denotes the k-th Grassmann angle of T F (B (n) ). Thus, relation (2.8) yields where the last equality follows from Theorem 3.3.
In the A-case the construction is analogous. Let 0 < j ≤ n and let a 1 , . . . , a j be positive numbers. Select a random and uniform subset {i 1 , . . . , i j−1 } ⊆ {1, . . . , n −1} with 1 ≤ i 1 < · · · < i j−1 ≤ n − 1, put i j := n, and define the corresponding j-face which is uniformly distributed in F j (A (n) ). We define the random signal x = (x 1 , . . . , x n ) by x m = l:i l ≥m a l for m = 1, . . . , n. Then, x belongs to the relative interior of F A (i 1 , . . . , i j−1 ). This yields the following proposition, which is analogous to the B-case and can be proven in a similar way.

Proposition 3.18
Let 0 < j ≤ k ≤ n and x ∈ R n be a random signal constructed as above. If G : R n → R k is a Gaussian random matrix, then it holds that

Schläfli Orthoschemes
Again, we start with the B-case. Let 0 ≤ j ≤ n be given. Take j + 1 positive numbers a 0 , a 1 , . . . , a j satisfying a 0 + · · · + a j = 1 and select a random and uniform subset Consider the signal x = (x 1 , . . . , x n ) given by By construction, we have x ∈ relint S B (i 0 , . . . , i j ) and S B (i 0 , . . . , i j ) is random and uniformly distributed on F j (K B n ). Then, the B-case of Theorem 3.1 yields the following proposition.

Proposition 3.19
Let 0 ≤ j ≤ k ≤ n and let x ∈ R n be a random signal constructed as above. If G : R n → R k is a Gaussian random matrix, then it holds that In the A-case, recall the definition of the Schläfli orthoscheme of type A: Take 0 ≤ j ≤ n and positive numbers a 1 , . . . , a j+1 satisfying a 1 + · · · + a j+1 = 1. Also, select a random and uniform subset {i 1 , . . . , i j+1 } ⊆ {1, . . . , n + 1} such that 1 ≤ i 1 < · · · < i j+1 ≤ n + 1. If i j+1 = n + 1, we define the corresponding j-face of K A n to be (which corresponds to the type of face described in (3.2)), while for i j ≤ n we define (which corresponds to the type of face described in (3.3)). Due to the one-to-one correspondence between the collections of indices 1 ≤ i 1 < · · · < i j+1 ≤ n + 1 and the j-faces of K A n , the face S A (i 1 , . . . , i j+1 ) is uniformly distributed on the set F j (K A n ) of all j-faces. We consider the random signal x = (x 1 , . . . , x n+1 ) given by where c is chosen to ensure that x 1 + · · · + x n+1 = 0. It can be easily checked that the signal x belongs to the relative interior of S A (i 1 , . . . , i j ). Then, the A-case of Theorem 3.1 yields the following proposition.

Proposition 3.20
Let 0 ≤ j ≤ k ≤ n and x ∈ R n+1 be a random signal constructed as above. If G : R n+1 → R k is a Gaussian random matrix, then it holds that

Proofs: Angle Sums of Weyl Chambers and Schläfli Orthoschemes
In this section, we present the proofs of Propositions 3.8, 3.7, and Theorem 3.5. Most of the proofs rely on explicit expressions for the conic intrinsic volumes of the Weyl chambers. Recall that the Weyl chambers A (n) and B (n) are defined by

Proof of Proposition 3.8
Let Recall that for l = (l 1 , . . . , l j+b ) such that l 1 , . . . , l j ∈ N, l j+1 , . . . , l j+b ∈ N 0 , and l 1 + · · · + l j+b = n, we define Our goal is to show that for all k ∈ {0, . . . , n}, Proof of Proposition 3.8 Let ( j, b) ∈ N 2 0 \ {(0, 0)} and k ∈ {0, . . . , n} be given. By the product formula for conic intrinsic volumes (2.5), the generating polynomial of the intrinsic volumes of T l can be written as We can consider each sum on the right-hand side separately. Using the representations of Stirling numbers of the first kind and their B-analogues from (2.9) and (2.11), as well as the intrinsic volumes of the Weyl chambers stated in (4.2), we obtain Note that for i = 0 we put (t + 1)(t + 3) · . . . · (t + 2i − 1) := 1 by convention, which is consistent with υ 0 ({0}) = 1. This yields the following formula: where t r := t (t + 1) · . . . · (t + r − 1), r ∈ N, denotes the rising factorial. Thus, the k-th conic intrinsic volume of T l is the coefficient of t k in the above polynomial P T l (t). Note that this already implies ν k (T l ) = 0 for k < j. Thus, the left-hand side of (4.3) is 0, which coincides with the right-hand side, since for k < j we have Therefore, we only need to consider the case k ≥ j. Let P (n) l 1 ,...,l j+b (m), where m = 0, . . . , n, be the coefficients of the polynomial Using the notation just introduced, we obtain Then, we can observe that Thus, to prove the proposition, it suffices to show that for all ( j, b) ∈ N 2 0 \ {(0, 0)} and k ∈ { j, . . . , n} we have To this end, we can introduce a new variable x and write, by expanding the product, Using (2.9) and the exponential generating function in two variables for the Stirling numbers of the first kind stated in (2.12), we obtain From (2.11) and (2.13), we similarly get (4.7) Thus, we have where we set c = c(x) = log(1 − x). The exponential generating function of the b/2-Stirling numbers stated in (2.18) yields It follows that Furthermore, using (2.17) we obtain Taking all this into consideration, we obtain which coincides with (4.5) and therefore completes the proof.

Proof of Theorem 3.5
Let b ∈ N. Recall that K B = K B n 1 × · · · × K B n b and K A = K A n 1 × · · · × K A n b for n 1 , . . . , n b ∈ N 0 such that n := n 1 + · · · + n b , where for d ∈ N, denote the Schläfli orthoschemes of types B and A in R d , respectively R d+1 . For convenience, we set K B 0 = K A 0 = {0}. We want to show that holds for all j ∈ {0, . . . , n} and k ∈ {0, . . . , n}, where for d ∈ {0, 1/2, 1} we define Proof of Theorem 3. 5 We divide the proof into three steps. In the first step we describe the tangent cones T F (K B ) in terms of products of Weyl chambers. In Step 2, we derive a formula for the generalized angle sums of the T F (K B )'s and show that the derived formula simplifies to the desired constant R 1 (k, j, b, (n 1 , . . . , n b )). In the third step, following the arguments of the B-case, we write the conic intrinsic volumes of the tangent cones T F (K A ) in terms of products of Weyl chambers, counted with certain multiplicity, and show that the formula for the generalized angle sums in the B-case also holds in the A-case.
Step 1. Let j ∈ {0, . . . , n} and k ∈ {0, . . . , n} be given. It is easy to check that the constant R d (k, j, b, (n 1 , . . . , n b )) vanishes for k < j, so that we need to consider the case k ≥ j only. For each j-face F of K B , there are numbers j 1 , . . . , j b ∈ N 0 satisfying j 1 + · · · + j b = j, such that . Thus, the tangent cone of K B at F is given by the following product formula: In order to see this, observe that relint Then, we have Applying Lemma 3.9 to the individual terms in the product, we observe that the collection where F 1 ∈ F j 1 (K B n 1 ), . . . , F b ∈ F j b (K B n b ), coincides (up to isometries) with the collection of cones (1) 0 + · · · + i (1) j 1 +1 = n 1 , . . . , i

This yields
(4.8) Step 2. Similarly to the proof of Proposition 3.8, we observe that ν k (G i ) is the coefficient of t k in the polynomial (1) (4.9) Following the same arguments as in the proof of Proposition 3.8, we obtain that term in (4.9).
We introduce new variables x 1 , . . . , x b and expand the product to write the right-hand side as follows: Using the formulas (4.6) and (4.7) we arrive at (4.10) By introducing a new variable u and expanding the product again, we arrive at k, j, b, (n 1 , . . . , n b )), which completes the proof of the B-case.
Step 3. In the A-case, instead of considering K A , we look at K A = K A n 1 × · · · × K A n b for n 1 , . . . , n b ∈ N 0 such that n := n 1 + · · · + n b , where denotes the unbounded polyhedral set related to K A d . Note that K A 0 = R. Following the arguments of Step 1, we can write the tangent cones of T F (K A ) (and also T F ( K A )) as products of tangent cones at their respective faces. Using υ k (K A n ) = υ k+1 (K A n ⊕ L n+1 ) and (3.1), we obtain which coincides with (4.10) and therefore completes the proof. Note that we used (4.6) and which follows from the binomial series.

Proof of Proposition 3.7
At first we show the formula for K B . In (4.8) we saw that holds true for all j, k ∈ {0, . . . , n}, where