Projections and angle sums of permutohedra and other polytopes
aa r X i v : . [ m a t h . M G ] S e p PROJECTIONS AND ANGLE SUMS OF PERMUTOHEDRA AND OTHERPOLYTOPES
THOMAS GODLAND AND ZAKHAR KABLUCHKO
Abstract.
Let ( x , . . . , x n ) ∈ R n . Permutohedra of types A and B are convex polytopes in R n defined by P An = conv { ( x σ (1) , . . . , x σ ( n ) ) : σ ∈ Sym( n ) } and P Bn = conv { ( ε x σ (1) , . . . , ε n x σ ( n ) ) : ( ε , . . . , ε n ) ∈ {± } n , σ ∈ Sym( n ) } , where Sym( n ) denotes the group of permutations of the set { , . . . , n } . We derive a closed formulafor the number of j -faces of G P An and G P Bn for a linear map G : R n → R d satisfying some minorgeneral position assumptions. In particular, we will show that the face numbers of the projectedpermutohedra do not depend on the linear mapping G . Furthermore, we derive formulas for thesum of the conic intrinsic volumes of the tangent cones of P An and P Bn at all of their j -faces. Thesame is done for the Grassmann angles. We generalize all these results to polytopes whose normalfan is the fan of some hyperplane arrangement. Introduction
In the work of Donoho and Tanner [8] the following interesting statement can be found: Thenumber of j -faces of the image of the n -dimensional cube [0 , n under a linear map G : R n → R d does not depend on the choice of G provided G is in “general position”. That is to say, the cube isan equiprojective polytope as defined by Shephard [23]. More precisely, by [8, Eq. (1.6)] we have f j ( G [0 , n ) = 2 (cid:18) nj (cid:19) n − j − X l = n − d (cid:18) n − j − l (cid:19) ; (1.1)for all 0 ≤ j < d ≤ n , where f j ( P ) denotes the number of j -faces of a polytope P , and G [0 , n is theimage of [0 , n under G . In the present paper, we investigate similar questions for the permutohedra of types A and B . These are the polytopes P An and P Bn in R n defined by P An := P An ( x , . . . , x n ) := conv { ( x σ (1) , . . . , x σ ( n ) ) : σ ∈ Sym( n ) (cid:9) and P Bn := P Bn ( x , . . . , x n ) := conv { ( ε x σ (1) , . . . , ε n x σ ( n ) ) : ( ε , . . . , ε n ) ∈ {± } n , σ ∈ Sym( n ) (cid:9) Mathematics Subject Classification.
Primary: 52A22, 60D05. Secondary: 11B73, 51F15, 52B05, 52B11,52A55.
Key words and phrases.
Permutohedra, polytopes, f -vector, projections, normal fans, polyhedral cones, conicintrinsic volumes, Grassmann angles, Stirling numbers, hyperplane arrangements, Weyl chambers, reflection arrange-ments, characteristic polynomials, zonotopes.TG and ZK acknowledge support by the German Research Foundation under Germany’s Excellence Strategy EXC2044 – 390685587, Mathematics M¨unster: Dynamics - Geometry - Structure. for a point ( x , . . . , x n ) ∈ R n . Here, Sym( n ) denotes the group of all permutations of the set { , . . . , n } . These polytopes have been studied starting with the work of Schoute [22] in 1911;see [6, 19, 12] as well as [29, Example 0.10], [28, Section 5.3], [4, pp. 58–60, 254–258] and [7,Example 2.2.5]. We prove that under certain minor general position assumptions on the linear map G : R n → R d , the number of j -faces of the projected permutohedra G P An and G P Bn is constant andgiven by the formulas f j ( G P An ) = 2 (cid:26) nn − j (cid:27) (cid:18)(cid:20) n − jn − d + 1 (cid:21) + (cid:20) n − jn − d + 3 (cid:21) + . . . (cid:19) (1.2)for all 1 ≤ j < d ≤ n −
1, provided that x > . . . > x n , and f j ( G P Bn ) = 2 T ( n, n − j ) (cid:0) B ( n − j, n − d + 1) + B ( n − j, n − d + 3) + . . . (cid:1) , (1.3)for all 1 ≤ j < d ≤ n , provided that x > . . . > x n >
0. Here, (cid:2) nk (cid:3) and (cid:8) nk (cid:9) denote the Stirlingnumbers of the first and second kind, respectively. The Stirling number of the first kind (cid:2) nk (cid:3) canbe defined as the number of permutations of the set { , . . . , n } having exactly k cycles, while theStirling number of the second kind (cid:8) nk (cid:9) is defined as the number of partitions of the set { , . . . , n } into k non-empty, disjoint subsets. The numbers B ( n, k ) and T ( n, k ) denote the B -analogues tothe Stirling numbers of the first and second kind, respectively, defined by the following formulas:( t + 1)( t + 3) · . . . · ( t + 2 n −
1) = n X k =0 B ( n, k ) t k , T ( n, k ) = n X m = k m − k (cid:18) nm (cid:19)(cid:26) mk (cid:27) . It turns out that these results can be generalized to any polytope P ⊂ R n whose normal fan,that is the set of normal cones at all faces F of P , coincides with the fan of some linear hyperplanearrangement A . A linear hyperplane arrangement is a finite collection of distinct linear hyperplanesin R n . The hyperplanes dissect the space into finitely many cones or chambers, and the fan of thearrangement is the set of all faces of these chambers. The number of j -faces of the projectedpolytope GP does not depend on the choice of the linear mapping G (under some minor generalposition assumptions on G ) and can be expressed in terms of the coefficients of the ( n − j )-th levelcharacteristic polynomials of the hyperplane arrangement A .It is known [29, Theorem 7.16] that zonotopes, that is Minkowski sums of finitely many linesegments, are special cases of the named class of polytopes. One simple special example is the cube[0 , n appearing in the formula (1.1). We show that the permutohedra of types A and B are alsospecial cases of this class of polytopes since their faces and normal fans can be characterized interms of reflection arrangements of types A n − and B n , respectively. On the other hand, it turnsout that permutohedra are zonotopes only in some rare exceptional cases, namely if the numbers x , . . . , x n form an arithmetic sequence.From the formulas (1.2) and (1.3), we derive results on generalized angle sums of permutohedra.In particular, we compute the sum of the d -th conic intrinsic volumes υ d of the tangent cones T F of P An and P Bn at their j -faces F as follows: X F ∈F j ( P An ) υ d ( T F ( P An )) = (cid:26) nn − j (cid:27)(cid:20) n − jn − d (cid:21) , for all 0 ≤ j ≤ d ≤ n − , X F ∈F j ( P Bn ) υ d ( T F ( P Bn )) = T ( n, n − j ) B ( n − j, n − d ) , for all 0 ≤ j ≤ d ≤ n. ROJECTIONS AND ANGLE SUMS OF PERMUTOHEDRA AND OTHER POLYTOPES 3
The same is done for the sums of the d -th Grassmann angles γ d of the tangent cones. The corre-sponding formulas are X F ∈F j ( P An ) γ d ( T F ( P An )) = 2 (cid:26) nn − j (cid:27) ∞ X l =0 (cid:20) n − jn − d − l − (cid:21) , for all 0 ≤ j ≤ d ≤ n − , X F ∈F j ( P Bn ) γ d ( T F ( P Bn )) = 2 T ( n, n − j ) ∞ X l =0 B ( n − j, n − d − l − , for all 0 ≤ j ≤ d ≤ n. Moreover, we compute the same angle sums for the above-mentioned more general class of polytopeswhose normal fans are fans of hyperplane arrangements.The paper is organized as follows. Section 2 introduces the necessary notation and some im-portant but well-known definitions and results from convex and integral geometry. In Section 3, westate the main results of this paper. In its first part, Section 3.1, we recall various characterizationsof the permutohedra, their faces, and normal fans. Section 3.2 contains some necessary results ongeneral position. In Section 3.3, we state the formulas for the number of faces of the projectedpermutohedra and more general polytopes mentioned above. Finally, Section 3.4 deals with theangle sums of these polytopes. Section 4 is dedicated to the proofs of the results from Section 3.2.
Preliminaries
In this section, we are going to introduce necessary facts and notation from convex and integralgeometry. These facts are well-known and can be skipped at first reading.2.1.
Facts from convex geometry.
For a set M ⊂ R n denote by lin M and aff M the linear hulland the affine hull of M , respectively. They are defined as the minimal linear and the minimalaffine subspace of R n containing M , respectively. Equivalently, lin M can be defined as the setof all linear combinations of elements in M , while aff M can be defined as the set of all affinecombinations of elements in M . Similarly, the convex hull of M is denoted by conv M and definedas the minimal convex set containing M , or, equivalently,conv M := (cid:8) λ x + . . . + λ m x m : m ∈ N , x , . . . , x m ∈ M, λ + . . . + λ m ≥ , λ + . . . + λ m = 1 (cid:9) . The positive hull of a set M is denoted by pos M and defined aspos M := (cid:8) λ x + . . . + λ m x m : m ∈ N , x , . . . , x m ∈ M, λ , . . . , λ m ≥ (cid:9) . The relative interior of a set M is the set of all interior points of M relative to its affine hull aff M and it is denoted by relint M . The set of interior points of M is denoted by int M .A polyhedral set is an intersection of finitely many closed half-spaces (whose boundaries neednot pass through the origin). A bounded polyhedral set is called polytope . Equivalently, a polytopecan be defined as the convex hull of a finite set of points. A polyhedral cone (or just cone) is anintersection of finitely many closed half-spaces whose boundaries contain the origin and thereforea special case of the polyhedral sets. Equivalently, a polyhedral cone can be defined as the positivehull of a finite set of points. The dimension of a polyhedral set P is defined as the dimension of itsaffine hull aff P .A supporting hyperplane for a polyhedral set P ⊂ R n is an affine hyperplane H with theproperty that H ∩ P = ∅ and P lies entirely in one of the closed half-spaces bounded by H .A face of a polyhedral set P (of arbitrary dimension) is a set of the form F = P ∩ H , for asupporting hyperplane H , or the set P itself. Equivalently, the faces of a polyhedral set P areobtained by replacing some of the half-spaces, whose intersection defines the polyhedral set, by THOMAS GODLAND AND ZAKHAR KABLUCHKO their boundaries and taking the intersection. The set of all faces of P is denoted by F ( C ) and theset of all k -dimensional faces (or just k -faces) of P by F k ( P ) for k ∈ { , . . . , n } . The number of k -faces of P is denoted by f k ( C ) := F k ( C ). In general, the number of elements in a set M isdenoted by | M | or M . The 0-dimensional faces are called vertices . In the case of a cone, the onlypossible vertex is the origin.The dual cone C ◦ (or polar cone) of a cone C ⊂ R n is defined as C ◦ := { v ∈ R n : h v, x i ≤ ∀ x ∈ C } . The tangent cone T F ( P ) of a polyhedral set P ⊂ R n at a face F of P is defined by T F ( P ) = { x ∈ R n : f + εx ∈ P for some ε > } , where f is any point in the relative interior of F . This definition does not depend on the choiceof f . The normal cone of P at the face F is defined as the dual of the tangent cone, that is N F ( P ) = T F ( P ) ◦ . Grassmann angles and conic intrinsic volumes.
Now, we are going to introduce someimportant geometric functionals of cones. Let C ⊂ R n be a cone and g be an n -dimensionalstandard Gaussian distributed vector. Then, the k -th conic intrinsic volume of C is defined as υ k ( C ) := X F ∈F k ( C ) P (Π C ( g ) ∈ relint F ) , k = 0 , . . . , n, where Π C denotes the orthogonal projection on C , that is, Π C ( x ) is the vector in C which minimizesthe Euclidean distance to x ∈ R n .The conic intrinsic volumes are the analogues of the usual intrinsic volumes in the setting ofconical or spherical geometry. Equivalently, the conic intrinsic volumes can be defined using thespherical Steiner formula, as done in [21, Section 6.5]. For further properties of conic intrinsicvolumes we refer to [2, Section 2.2] and [21, Section 6.5].Following Grnbaum [11], we define the Grassmann angles γ k ( C ), k ∈ { , . . . , n } , of cone C asfollows. Let W n − k be random linear subspace of R n with uniform distribution on the Grassmannianof all ( n − k )-dimensional subspaces. Then, the k -th Grassmann angle of C is defined as γ k ( C ) := P ( W n − k ∩ C = { } ) , k = 0 , . . . , n. If the lineality space C ∩ − C of a cone C , which is the maximal linear subspace contained in C ,has dimension j ∈ { , . . . , n − } , the Grassmann angles satisfy1 = γ ( C ) = . . . = γ j ( C ) ≥ γ j +1 ( C ) ≥ . . . ≥ γ n ( C ) = 0 . As proved in [11, Eq. (2.5)], the Grassmann angles do not depend on the dimension of the ambientlinear subspace. This means that if we embed C in R N with N ≥ n , we obtain the same Grassmannangles. Therefore, it is convenient to write γ N ( C ) := 0 for all N ≥ dim C . If C is not a linearsubspace, then γ k ( C ) is also known as the k -th conic quermassintegral ; see [21, Eqs. (1)-(4)]or [13].The conic intrinsic volumes and Grassmann angles satisfy a linear relation, called the conicCrofton formula . More precisely, we have γ k ( C ) = 2 X i =1 , , ,... υ k + i ( C ) (2.1) ROJECTIONS AND ANGLE SUMS OF PERMUTOHEDRA AND OTHER POLYTOPES 5 for all k ∈ { , . . . , n } and for every cone C ⊂ R n which is not a linear subspace, according to [21,p.261]. Consequently, υ k ( C ) = 12 γ k − ( C ) − γ k +1 ( C ) , (2.2)for all k ∈ { , . . . , n } , where in the cases k = 0 and k = n we have to define γ − ( C ) = 1 and γ n +1 ( C ) = 0. Then, (2.2) follows from (2.1) and the identity υ ( C ) + υ ( C ) + . . . = 1 /
2; see [3,Eq. (5.3)]. 3.
Main results
Permutohedra of types A and B . In this section, we introduce permutohedra of types A and B , give an exact characterization of their faces, and compute their normal fans. Definitions of the permutohedra.
For x , . . . , x n ∈ R the permutohedron of type A is defined as thefollowing polytope in R n : P An = P An ( x , . . . , x n ) := conv (cid:8) ( x σ (1) , . . . , x σ ( n ) ) : σ ∈ Sym( n ) (cid:9) , where Sym( n ) is the group of all permutations of the set { , . . . , n } . The permutohedron lies inthe hyperplane { t ∈ R n : t + . . . + t n = x + . . . + x n } and therefore has at most dimension n − permutohedron of type B is defined as the following polytope in R n : P Bn = P Bn ( x , . . . , x n ) := conv (cid:8) ( ε x σ (1) , . . . , ε n x σ ( n ) ) : ε = ( ε , . . . , ε n ) ∈ {± } n , σ ∈ Sym( n ) (cid:9) . Note that P An ( x , . . . , x n ) does not change under permutations of x , . . . , x n , whereas P Bn ( x , . . . , x n )stays invariant under signed permutations. Therefore, it is not a restriction of generality to assumethat x ≥ . . . ≥ x n in the A -case and x ≥ . . . ≥ x n ≥ B -case.The next lemma is due to Rado [20] (see also [28, Section 5.3], [4, p. 257] and [17, CorollaryB.3]) and describes P An as a set of solutions to a finite system of affine inequalities. Lemma 3.1.
Assume that x ≥ . . . ≥ x n . Then, a point ( t , . . . , t n ) ∈ R n belongs to the permuto-hedron P An ( x , . . . , x n ) of type A if and only if t + · · · + t n = x + . . . + x n and, for every non-empty subset M ⊂ { , . . . , n } , we have X i ∈ M t i ≤ x + . . . + x | M | . An analogous result for the permutohedron of type B , together with a proof and references tothe original literature, can be found in [17, Corollary C.5.a]. Lemma 3.2.
Assume that x ≥ . . . ≥ x n ≥ . Then, a point ( t , . . . , t n ) ∈ R n belongs to thepermutohedron P Bn ( x , . . . , x n ) of type B if and only if for every non-empty subset M ⊂ { , . . . , n } ,we have X i ∈ M | t i | ≤ x + . . . + x | M | . (3.1) THOMAS GODLAND AND ZAKHAR KABLUCHKO
Faces of the permutohedra.
We now state an exact characterization of the faces of both types ofpermutohedra. To this end, we need to introduce some useful notation.Let R n,j be the set of all ordered partitions ( B , . . . , B j ) of the set { , . . . , n } into j non-empty, disjoint and distinguishable subsets B , . . . , B j . Furthermore, let T n,j be the set of all pairs( B , η ), where B = ( B , . . . , B j +1 ) is an ordered partition of the set { , . . . , n } into j + 1 disjointdistinguishable subsets such that B , . . . , B j are non-empty, whereas B j +1 may be empty or not,and η : B ∪ . . . ∪ B j → {± } . In what follows, we write η i := η ( i ) for ease of notation. Proposition 3.3.
Suppose that x > . . . > x n . Then, for j ∈ { , . . . , n − } , the j -dimensionalfaces of P An ( x , . . . , x n ) are in one-to-one correspondence with the ordered partitions B ∈ R n,n − j .The j -face corresponding to the ordered partition B = ( B , . . . , B n − j ) ∈ R n,n − j is given by F B = conv { ( x σ (1) , . . . , x σ ( n ) ) : σ ∈ I B } . Here, I B ⊂ Sym ( n ) is the set of all permutations σ ∈ Sym ( n ) such that σ ( B ) = { , . . . , | B |} , σ ( B ) = {| B | + 1 , . . . , | B ∪ B |} , . . . ,σ ( B n − j ) = {| B ∪ . . . ∪ B n − j − | + 1 , . . . , n } . Equivalently, the face F B can be written as F B = (cid:26) ( t , . . . , t n ) ∈ P An ( x , . . . , x n ) : X i ∈ B ∪ ... ∪ B l t i = x + . . . + x | B ∪ ... ∪ B l | ∀ l = 1 , . . . , n − j − (cid:27) . Proposition 3.4.
Suppose that x > . . . > x n > . Then, for j ∈ { , . . . , n } , the j -dimensionalfaces of P Bn ( x , . . . , x n ) are in one-to-one correspondence with the pairs ( B , η ) ∈ T n,n − j . The j -facecorresponding to the pair ( B , η ) , where B = ( B , . . . , B n − j +1 ) , is given by F B ,η = conv { ( ε x σ (1) , . . . , ε n x σ ( n ) ) : ( σ, ε ) ∈ I B ,η } . Here, I B ,η ⊂ Sym( n ) × {± } n is the set of all pairs ( σ, ε ) ∈ Sym( n ) × {± } n such that σ ( B ) = { , . . . , | B |} , σ ( B ) = {| B | + 1 , . . . , | B ∪ B |} , . . . ,σ ( B n − j +1 ) = {| B ∪ . . . ∪ B n − j | + 1 , . . . , n } and ε i = η i for all i ∈ B ∪ . . . ∪ B n − j , while the remaining ε i ’s take arbitrary values in the set {± } . Equivalently, the face F B ,η can be written as F B ,η = (cid:26) ( t , . . . , t n ) ∈ P Bn ( x , . . . , x n ) : X i ∈ B ∪ ... ∪ B l η i t i = x + . . . + x | B ∪ ... ∪ B l | ∀ l = 1 , . . . , n − j (cid:27) . Proofs of Proposition 3.3 can be found in [4, pp. 254-256] or in [28, Section 5.3]. Without proof,versions of the same proposition are stated in [19, Proposition 2.6] and in Exercise 2.9 on p. 96of [7]. For completeness, a proof of Proposition 3.4 (which may also be known) will be provided inSection 4.1.
Normal fans of permutohedra.
Following [29, Chapter 7], a fan in R n is defined as a family F ofnon-empty cones with the following two properties:(i) Every non-empty face of a cone in F is also a cone in F .(ii) The intersection of any two cones in F is a face of both. ROJECTIONS AND ANGLE SUMS OF PERMUTOHEDRA AND OTHER POLYTOPES 7
For a non-empty polytope P ⊂ R n the normal fan of P is defined as the set of normal cones of P ,that is N ( P ) := { N F ( P ) : F ∈ F ( P ) } . For a hyperplane arrangement A , which is a finite set of linear hyperplanes in R n , the comple-ment R n \ S H ∈A H is a disjoint union of open convex sets. The set of closures of these “regions”is denoted by R ( A ) and called the conical mosaic generated by A . Clearly, the set of all faces ofthese cones defines a fan denoted by F ( A ): F ( A ) = [ C ∈R ( A ) F ( C ) . Also, we denote the set of j -dimensional cones of this fan by F j ( A ).The following theorems exactly characterize the normal fans of the permutohedra of types A and B . Denote by A ( A n − ) the hyperplane arrangement consisting of the hyperplanes { β ∈ R n : β i = β j } , ≤ i < j ≤ n. (3.2)Similarly, let A ( B n ) be the hyperplane arrangement that consists of the hyperplanes { β ∈ R n : β i = β j } , ≤ i < j ≤ n, { β ∈ R n : β i = − β j } , ≤ i < j ≤ n, (3.3) { β ∈ R n : β i = 0 } , ≤ i ≤ n. These arrangements, also called reflection arrangements of types A n − and B n , as well as the conesthey generate (called the Weyl chambers), will be further discussed in Section 4.3. Theorem 3.5.
For x > . . . > x n the normal fan N ( P An ( x , . . . , x n )) of the permutohedron of type A coincides with the fan F ( A ( A n − )) generated by the hyperplane arrangement A ( A n − ) . Theorem 3.6.
For x > . . . > x n > the normal fan N ( P Bn ( x , . . . , x n )) of the permutohedron oftype B coincides with the fan F ( A ( B n )) generated by the hyperplane arrangement A ( B n ) . Both theorems seem to be known, see, e.g., [12, Section 3.1], but for the sake of completenesswe will give their proofs in Section 4.1. For example, the normal cones at the vertices of thepermutohedra coincide with the Weyl chambers of types A n − and B n , which was used to computetheir statistical dimension in [3, Proposition 3.5].3.2. Results on general position.
Before stating our main results on the face numbers of pro-jected permutohedra, we need to introduce the terminology of general position in the context ofhyperplane arrangements and polyhedral sets. Moreover, we formulate general position assumptionsfor both types of permuthohedra that are necessary for our main results in Section 3.3.Let M be an affine subspace of R n . Denote by L ⊂ R n the unique linear subspace such that M = t + L holds for some t ∈ R n , that is the translation of M passing through the origin. We saythat M is in general position with respect to a linear subspace L ′ ⊂ R n ifdim( L ∩ L ′ ) = max(dim L + dim L ′ − n, . A linear subspace L ′ is said to be in general position with respect to a polyhedral set P if the affinehull of each face F of P is in general position with respect to L ′ .For a linear hyperplane arrangement A in R n , the lattice L ( A ) generated by A consists of alllinear subspaces of R n that can be represented as intersections of hyperplanes from A . Denote by THOMAS GODLAND AND ZAKHAR KABLUCHKO L j ( A ) the set of j -dimensional subspaces from the lattice L ( A ). A linear subspace L ′ ⊂ R n is saidto be in general position with respect to the hyperplane arrangement A , if all K ∈ L ( A ) satisfydim( K ∩ L ′ ) = max(dim L ′ + dim K − n, , (3.4)that is, if L ′ is in general position with respect to each K ∈ L ( A ).Now, we are able to formulate two equivalent general position assumptions that we need toimpose on a linear mapping G ∈ R d × n in the case of a permutohedron of type A . Corollary 3.7.
Let ≤ d ≤ n − and x > . . . > x n . For a matrix G ∈ R d × n with rank G = d ,the following two conditions are equivalent:(A1) The ( n − d ) -dimensional linear subspace ker G is in general position with respect to P An ( x , . . . , x n ) .(A2) The d -dimensional linear subspace (ker G ) ⊥ is in general position with respect to thereflection arrangement A ( A n − ) defined in (3.2) . An analogous result can be formulated for the permutohedron of type B . Corollary 3.8.
Let ≤ d ≤ n and x > . . . > x n > . For a matrix G ∈ R d × n with rank G = d ,the following two conditions are equivalent:(B1) The ( n − d ) -dimensional linear subspace ker G is in general position with respect to P Bn ( x , . . . , x n ) .(B2) The d -dimensional linear subspace (ker G ) ⊥ is in general position with respect to thereflection arrangement A ( B n ) defined in (3.3) . Corollaries 3.7 and 3.8 follow from the more general Theorem 3.11, which we will state inSection 3.3. Therefore, their proofs will be postponed to Section 4.2.3.3.
Face numbers of projected permutohedra and more general polytopes.
In this sec-tion, we state our main results on the number of faces of projected permutohedra of types A and B and, more generally, of polytopes whose normal fan coincides with the fan generated by a hyper-plane arrangement. In case of the named polytopes, the face numbers of the projected polytopesare independent of the projection provided it satisfies some minor general position assumption. Permutohedra of types A and B . The formulas will be stated in terms of Stirling numbers definedas follows. The (signless)
Stirling number of the first kind (cid:2) nk (cid:3) is the number of permutations of theset { , . . . , n } having exactly k cycles. Equivalently, these numbers can be defined as the coefficientsof the polynomial t ( t + 1) · . . . · ( t + n −
1) = n X k =0 (cid:20) nk (cid:21) t k (3.5)for n ∈ N , with the convention that (cid:2) nk (cid:3) = 0 for n ∈ N , k / ∈ { , . . . , n } and (cid:2) (cid:3) = 1. The B -analogues to the Stirling numbers of the first kind, denoted by B ( n, k ), are defined as the coefficientsof the polynomial ( t + 1)( t + 3) · . . . · ( t + 2 n −
1) = n X k =0 B ( n, k ) t k (3.6)for n ∈ N and, by convention, B ( n, k ) = 0 for k / ∈ { , . . . , n } . The triangular array of integers B ( n, k ) appears as entry A028338 in [24]. ROJECTIONS AND ANGLE SUMS OF PERMUTOHEDRA AND OTHER POLYTOPES 9
The
Stirling number of the second kind (cid:8) nk (cid:9) is defined as the number of partitions of the set { , . . . , n } into k non-empty subsets. The B -analogues to the Stirling numbers of the second kind,denoted by T ( n, k ), are defined as T ( n, k ) = n X m = k m − k (cid:18) nm (cid:19)(cid:26) mk (cid:27) . (3.7)They appear as entry A039755 in [24] and were studied by Suter [27].Our main results for the permutohedra of types A and B are as follows. Theorem 3.9.
Let x > . . . > x n be given. For a matrix G ∈ R d × n with rank G = d and satisfyingone of the equivalent general position assumptions (A1) or (A2), we have f j ( G P An ) = 2 (cid:26) nn − j (cid:27) (cid:18)(cid:20) n − jn − d + 1 (cid:21) + (cid:20) n − jn − d + 3 (cid:21) + . . . (cid:19) , for all ≤ j < d ≤ n − . Theorem 3.10.
Let x > . . . > x n > be given. For a matrix G ∈ R d × n with rank G = d satisfyingone of the equivalent general position assumptions (B1) or (B2), we have f j ( G P Bn ) = 2 T ( n, n − j ) (cid:0) B ( n − j, n − d + 1) + B ( n − j, n − d + 3) + . . . (cid:1) , for all ≤ j < d ≤ n . The proofs of Theorems 3.9 and 3.10 are postponed to Section 4.3.
A more general class of polytopes.
We are able to formulate a more general result which is validfor all polytopes P ⊂ R n whose normal fan N ( P ) := { N F ( P ) : F ∈ F ( P ) } coincides with thefan F ( A ) := S C ∈R ( A ) F ( C ) generated by some hyperplane arrangement A . Here, R ( A ) denotesthe conical mosaic in R n consisting of the n -dimensional cones generated by A . In Theorems 3.5and 3.6 we already observed that the permutohedra of types A and B are special cases of this classof polytopes. Before stating the result, we need to introduce the characteristic polynomial of ahyperplane arrangement.The rank of a linear hyperplane arrangement A in R n is defined byrank( A ) = n − dim (cid:18) \ H ∈A H (cid:19) , rank( ∅ ) = 0 . The characteristic polynomial χ A ( t ) of A can be defined by the following Whitney formula: χ A ( t ) = X C⊂A ( − C t n − rank( C ) ; (3.8)see, e.g., [18, Lemma 2.3.8] or [25, Theorem 2.4], as well as [25, Section 1.3] or [26, Section 3.11.2]for other definitions using the M¨obius function on the intersection poset of A .Similar to the case of the permutohedra, we need to impose certain general position assump-tions on the linear mapping G under consideration. Theorem 3.11.
Let P ⊂ R n be a polytope such that the normal fan N ( P ) coincides with thefan F ( A ) of some hyperplane arrangement A . For ≤ d ≤ dim P and a matrix G ∈ R d × n with rank G = d the following two general position assumptions are equivalent:(G1) The ( n − d ) -dimensional linear subspace ker G is in general position with respect to P .(G2) The d -dimensional linear subspace (ker G ) ⊥ is in general position with respect to thehyperplane arrangement A . The proof of this theorem is postponed to Section 4.2.
Theorem 3.12.
Let P ⊂ R n be a polytope such that the normal fan N ( P ) coincides with the fan F ( A ) of a hyperplane arrangement A . Moreover, let G ∈ R d × n be a matrix with rank G = d suchthat one of the equivalent general position assumptions (G1) or (G2) is satisfied. Then, the numberof j -faces of the projected polytope GP is independent of the linear map G and given by f j ( GP ) = 2 X M ∈L n − j ( A ) ( a Mn − d +1 + a Mn − d +3 + . . . ) , for ≤ j < d ≤ dim P , where the numbers a Mk are (up to a sign) the coefficients of the characte-ristic polynomial of the induced hyperplane arrangement A| M := { H ∩ M : H ∈ A , M * H } in theambient space M : χ A | M ( t ) = n − j X k =0 ( − n − j − k a Mk t k . (3.9) Also recall that L n − j ( A ) is set of all ( n − j ) -dimensional linear subspaces that can be representedas intersections of hyperplanes from A . By convention, we have L n ( A ) := { R n } . The proof of Theorem 3.12 is postponed to Section 4.3. As a consequence, the polytopesconsidered in Theorem 3.12 belong to the class of equiprojective polytopes as defined in [23].
Permutohedra and zonotopes as special cases.
As mentioned above, the permutohedra are specialcases of the above class of polytopes since their normal fans coincide with the fans of reflectionarrangements. Thus, Theorems 3.9 and 3.10 can be derived from Theorem 3.12 using formulasfor the coefficients of the characteristic polynomials of the reflection arrangements or rather theirrestrictions to linear subspaces M ∈ L n − j ( A ( A n − )), respectively, M ∈ L n − j ( A ( B n )). Thesecoefficients were already computed by Amelunxen and Lotz [2, Lemma 6.5]. In Section 4.3 however,we will prove Theorems 3.9 and 3.10 using an equivalent approach that includes computing thenumber of faces of Weyl chambers that are non-trivially intersected by a linear subspace.Besides permutohedra, the zonotopes are also a special case of the above class of polytopeswhose normal fan is the fan of a hyperplane arrangement. A zonotope Z = Z ( V ) ⊂ R n is aMinkowski sum of a finite number of line segments, and therefore, can be written as Z ( V ) = [ − v , v ] + . . . + [ − v p , v p ] + z for some p ∈ N , a matrix V = ( v , . . . , v p ) ∈ R n × p and z ∈ R n . Following [29, Definition 7.13], azonotope Z = Z ( V ) can equivalently be defined as the image of a cube under an affine map, thatis, Z ( V ) := V [ − , +1] p + z = { V y + z : y ∈ [ − , +1] p } . In the book of Ziegler [29, Theorem 7.16] it is proved that for a zonotope Z = Z ( V ) ⊂ R n , thenormal fan N ( Z ) of Z coincides with the fan F ( A ) of the hyperplane arrangement A = A V := { H , . . . , H p } in R n , where H i := { x ∈ R n : h x, v i i = 0 } for i = 1 , . . . , p . It is known [29, Example 7.15] that P An ( n, n − , . . . , ,
1) is a zonotope and the natural question arises if the permutohedra of types A and B are zonotopes for all ( x , . . . , x n ) ∈ R n . The following proposition shows that this is not thecase. ROJECTIONS AND ANGLE SUMS OF PERMUTOHEDRA AND OTHER POLYTOPES 11
Proposition 3.13.
For x > . . . > x n , the permutohedron P An ( x , . . . , x n ) of type A is a zonotopeif and only if x , . . . , x n are in arithmetic progression, that is, x = a + ( n − b, x = a + ( n − b, . . . , x n − = a + b, x n = a (3.10) for some a ∈ R and b > .For x > . . . > x n > , the permutohedron P Bn ( x , . . . , x n ) of type B is a zonotope if and onlyif x , . . . , x n are in arithmetic progression, that is if (3.10) holds for some a > and b > . The proof is postponed to Section 4.1.3.4.
Angle sums of permutohedra, zonotopes and other polytopes.
The next theoremfollows from Theorem 3.9 using a well-known formula from Affentranger and Schneider [1] and willbe proved in Section 4.4. Recall that the Grassmann angles γ d and the conic intrinsic volumes υ d were defined in Section 2.2. Theorem 3.14.
Let x > . . . > x n be given. We have X F ∈F j ( P An ) υ d ( T F ( P An )) = (cid:26) nn − j (cid:27)(cid:20) n − jn − d (cid:21) (3.11) and X F ∈F j ( P An ) γ d ( T F ( P An )) = 2 (cid:26) nn − j (cid:27) ∞ X l =0 (cid:20) n − jn − d − l − (cid:21) (3.12) for ≤ j ≤ d ≤ n − . The following theorem is an analogue of Theorem 3.14 in the B -case. Theorem 3.15.
Let x > . . . > x n > be given. We have X F ∈F j ( P Bn ) υ d ( T F ( P Bn )) = T ( n, n − j ) B ( n − j, n − d ) (3.13) and X F ∈F j ( P Bn ) γ d ( T F ( P Bn )) = 2 T ( n, n − j ) ∞ X l =0 B ( n − j, n − d − l −
1) (3.14) for ≤ j ≤ d ≤ n . Note that (3.11) and (3.13) recover results from Amelunxen and Lotz [2]. Among other things,they derived a formula for the j -th level characteristic polynomial of the reflection arrangements A ( A n − ) and A ( B n ), see [2, Lemma 6.5], which yields the sums of the ( n − d )-th conic intrinsicvolumes over all j -dimensional regions from F ( A ( A n − )), respectively F ( A ( B n )), see [2, Theo-rem 6.1]. These formulas coincide with the sums computed in (3.11) and (3.13) (with j replacedby n − j ). This does not come as a surprise since the normal fans of the permutohedra are thefans of the corresponding reflection arrangements, following Theorems 3.5 and 3.6. The proofs ofTheorems 3.14 and 3.15 are postponed to Section 4.4.For more general polytopes whose normal fans are generated by a hyperplane arrangement,the following result holds. Theorem 3.16.
Let P ⊂ R n be a polytope whose normal fan N ( P ) coincides with the fan F ( A ) of a hyperplane arrangement A . Then, we have X F ∈F j ( P ) υ d ( T F ( P )) = X M ∈L n − j ( A ) a Mn − d and X F ∈F j ( P ) γ d ( T F ( P )) = 2 X M ∈L n − j ( A ) ( a Mn − d − + a Mn − d − + . . . ) for ≤ j ≤ d ≤ dim P . Recall that the numbers a Mk were defined in (3.9) and L j ( A ) denotes theset of j -dimensional subspaces from the lattice of A . By convention, we put L n ( A ) := { R n } . The proof of this theorem is also postponed to Section 4.4. Theorems 3.14 and 3.15 can bededuced from Theorem 3.16 since the normal fans of the permutohedra of types A and B coincidewith the fans F ( A ( A n − )) and F ( A ( B n )), respectively, following Theorems 3.5 and 3.6. Using theknown formulas [2, Lemma 6.5] for the coefficients of the j -th level characteristic polynomials yieldsthe results. In Section 4.4, we are going to prove Theorems 3.14 and 3.15 by using Theorems 3.9and 3.10 on the face numbers of projected permutohedra of types A and B , respectively, and aformula of Affentranger and Schneider [1]. Let us also mention that applying Theorem 3.16 to afull-dimensional zonotope P with d = n we recover a formula stated in [16, Theorem 12].4. Proofs
This section is dedicated to proving the main results from Section 3. In Section 4.1, we aregoing to prove the characterization of the faces and the normal fans of permutohedra. Section 4.2contains the proofs of the equivalences between the general position assumptions of Theorem 3.11and its Corollaries 3.7 and 3.8 from Section 3.2. Moreover, in Section 4.3, we prove the main resultsof this paper on the number of j -faces of the projected permutohedra and the more general classof polytopes. Finally, we will also prove the results on the angle sums of the same polytopes inSection 4.4.4.1. Permutohedra: Proofs of Propositions 3.4 and 3.13, and Theorems 3.5 and 3.6.
Before starting with the proofs, let us mention the well-known fact that the points ( x σ (1) , . . . , x σ ( n ) )are indeed vertices of P An for all σ ∈ Sym( n ). Similarly, the points ( ε x σ (1) , . . . , ε n x σ ( n ) ) are verticesof P Bn for all ε ∈ {± } n , σ ∈ Sym( n ). Let us explain this in the B -case. It suffices to prove theclaim for the point x = ( x , . . . , x n ) of P Bn , where x ≥ . . . ≥ x n ≥
0. Suppose that there are points y = ( y , . . . , y n ) ∈ P Bn and z = ( z , . . . , z n ) ∈ P Bn such that x = ( y + z ) /
2. By Lemma 3.2, wehave | y | ≤ x and | z | ≤ x . Thus, we have y = z = x . Given this, we can consider the secondcoordinate in the same way. Inductively, we obtain y i = z i = x i for all i = 1 , . . . , n , which meansthat ( x , . . . , x n ) is indeed a vertex of P Bn .In the case where x > . . . > x n , the permutohedron P An has dimension n −
1. Similarly, for x > . . . > x n >
0, the permutohedron P Bn of type B has dimension n . Faces of permutohedra of type B . We are going to prove Proposition 3.4. Suppose that x > . . . >x n >
0. Recall that T n,n − j denotes the set of all pairs ( B , η ), where B = ( B , . . . , B n − j +1 ) isan ordered partition of the set { , . . . , n } into n − j + 1 disjoint distinguishable subsets such that B , . . . , B n − j are non-empty, whereas B n − j +1 may be empty or not, and η : B ∪ . . . ∪ B n − j → {± } .For Proposition 3.4, we want to prove that there is a one-to-one correspondence between the j -faces ROJECTIONS AND ANGLE SUMS OF PERMUTOHEDRA AND OTHER POLYTOPES 13 of P Bn ( x , . . . , x n ) and the pairs ( B , η ) ∈ T n,n − j , and that the j -face corresponding to the pair ( B , η )is given by F B ,η = conv { ( ε x σ (1) , . . . , ε n x σ ( n ) ) : ( σ, ε ) ∈ I B ,η } (4.1)= (cid:26) t ∈ P Bn ( x , . . . , x n ) : X i ∈ B ∪ ... ∪ B l η i t i = x + . . . + x | B ∪ ... ∪ B l | ∀ l = 1 , . . . , n − j (cid:27) . (4.2)Here, I B ,η ⊂ Sym( n ) × {± } n is the set of all pairs ( σ, ε ) ∈ Sym( n ) × {± } n such that σ ( B ) = { , . . . , | B |} , σ ( B ) = {| B | + 1 , . . . , | B ∪ B |} , . . . and ε i = η i for i ∈ B ∪ . . . ∪ B n − j . Proof of Proposition 3.4.
Let F ∈ F ( P Bn ) be a face of P Bn ( x , . . . , x n ) with x > . . . > x n > F = P Bn , which means there is nothing to prove, or there is a supporting hyperplane H = { t ∈ R n : α t + . . . + α n t n = b } for some α = ( α , . . . , α n ) ∈ R n \{ } and b ∈ R such that H ∩ P Bn = F and P Bn ⊂ H − := { t ∈ R n : α t + . . . + α n t n ≤ b } . (4.3)Without loss of generality, we may assume that α ≥ . . . ≥ α n ≥ { , . . . , n } to ( α , . . . , α n ) and all other objects). Then, α = · · · = α i | {z } group 1 > α i +1 = . . . = α i | {z } group 2 > . . . > α i m − +1 = . . . = α i m | {z } group m > α i m +1 = . . . = α n = 0 | {z } group m + 1 , (4.4)for some m ∈ { , . . . , n } and 1 ≤ i < . . . < i m ≤ n . Note that for i m = n , no α i ’s are required tobe zero, which means that the last group is empty. Then, P Bn ⊂ H − implies that α ε x σ (1) + . . . + α n ε n x σ ( n ) ≤ b, for all ε = ( ε , . . . , ε n ) ∈ {± } n , σ ∈ Sym( n ) . The first equation of (4.3) implies that there is a pair ( σ ′ , ε ′ ) ∈ Sym( n ) × {± } n such that α ε ′ x σ ′ (1) + . . . + α i m ε ′ i m x σ ′ ( i m ) = α ε ′ x σ ′ (1) + . . . + α n ε ′ n x σ ′ ( n ) = b. Since the α i ’s and the x i ’s are non-increasing and non-negative, the swapping lemma (see, e.g., [4,p. 254]) states that α ε x σ (1) + . . . + α n ε n x σ ( n ) attains its maximal value if we choose ε i = +1 and σ ( i ) = i for all i ∈ { , . . . , n } . It follows that, in fact, we have α x + . . . + α n x n = b. (4.5)Denote the groups of indices appearing in (4.4) by B = { , . . . , i } , . . . , B m = { i m − + 1 , . . . , i m } , B m +1 = { i m + 1 , . . . , n } , (4.6)where B m +1 may or may not be empty. Defining in our case η i := 1 for all i ∈ { , . . . , i m } weobtain that, under (4.6), the set I B ,η consists of all pairs ( σ, ε ) ∈ Sym( n ) × {± } n such that σ ( B ) = B , . . . , σ ( B m ) = B m , σ ( B m +1 ) = B m +1 and ε i = η i for all i ∈ { , . . . , i m } . Consequently, from (4.5) and (4.4) it follows that α ε x σ (1) + . . . + α n ε n x σ ( n ) = b for all ( σ, ε ) ∈ I B ,η . (4.7)Furthermore, it follows from (4.5), (4.4) and the swapping lemma that α ε x σ (1) + . . . + α n ε n x σ ( n ) < b for all ( σ, ε ) ∈ (Sym( n ) × {± } n ) \ I B ,η . (4.8)Indeed, if ( σ, ε ) / ∈ I B ,η , then there is the possibility that we have a strictly negative term on theleft-hand side of (4.8) which means that we could make it strictly larger by changing the sign ofthis term. Thus, we can assume all these terms to be non-negative. Then, ( σ, ε ) / ∈ I B ,η implies that there is a pair of indices 1 ≤ i < j ≤ n such that α i > α j und x σ ( i ) < x σ ( j ) and we can apply theswapping lemma to strictly increase the left-hand side.According to (4.7) and (4.8), the vertices ( ε x σ (1) , . . . , ε n x σ ( n ) ) with ( σ, ε ) ∈ I B ,η are the onlyvertices of P Bn that belong to the supporting hyperplane H . It follows from [29, Proposition 2.3]that F is the convex hull of these vertices, that is F = F B ,η , where F B ,η := conv (cid:8) ( ε x σ (1) , . . . , ε n x σ ( n ) ) : ( σ, ε ) ∈ I B ,η (cid:9) . (4.9)Essentially the same argument shows that, conversely, a set of the form F B ,η is a face of P Bn . Atthe beginning, we applied a signed permutation to all objects including ( α , . . . , α n ) to achieve thatthe α i ’s are non-increasing and non-negative. Applying the inverse signed permutation proves thatthe faces of P Bn coincide with the sets of the form F B ,η as defined in (4.9), for some pair ( B , η ) ∈ T n,m .Furthermore, for two different pairs ( B ′ , η ′ ) , ( B ′′ , η ′′ ) we have I B ′ ,η ′ = I B ′′ ,η ′′ , which implies that thecorresponding sets F B ′ ,η ′ and F B ′′ ,η ′′ are different, since their sets of vertices are different. Finally,the polytope F B ,η , for ( B , η ) ∈ T n,m , is isometric to the direct product P A | B | × . . . × P A | B m | × P B | B m +1 | ,which follows from the description of the vertices of F B ,η . It follows that dim F B ,η = n − m .Now, we prove the equivalence of the representations (4.1) and (4.2). To this end, we takesome pair ( B , η ) ∈ T n,m , assuming without restriction of generality that B := { , . . . , i } , B := { i + 1 , . . . , i } , . . . , B m +1 := { i m + 1 , . . . , n } , where 1 ≤ i < . . . < i m ≤ n for some m ∈ { , . . . , n } , and η i = 1 for i ∈ { , . . . , i m } . Our goal isto prove that F B ,η = M , where M := (cid:26) t ∈ P Bn ( x , . . . , x n ) : X i ∈ B ∪ ... ∪ B l t i = x + . . . + x | B ∪ ... ∪ B l | ∀ l = 1 , . . . , m (cid:27) . The inclusion F B ,η ⊂ M holds trivially and we only need to prove that M ⊂ F B ,η .Let α = ( α , . . . , α n ) ∈ R n \{ } be such that condition (4.4) holds. The above argumentsshow that the hyperplane H = { t ∈ R n : α t + . . . + α n t n = b } with b := α x + . . . + α n x n is asupporting hyperplane of the face F B ,η , that is H ∩ P Bn = F B ,η and P Bn ⊂ H − := { t ∈ R n : α t + . . . + α n t n ≤ b } . Suppose now that there is some y / ∈ F B ,η such that y ∈ M ⊂ P Bn ⊂ H − . This already yields α y + . . . + α i m y i m = α y + . . . + α n y n < t = α x + . . . + α i m x i m , since y ∈ H − , but y / ∈ H . It follows that α i m ( y + . . . + y i m ) + m − X l =1 ( α i l − α i l +1 ) i l X i =1 y i = α y + . . . + α i m y i m < α x + . . . + α i m x i m = α i m ( x + . . . + x i m ) + m − X l =1 ( α i l − α i l +1 ) i l X i =1 x i , which is a contradiction to y ∈ M . This proves that both representations (4.1) and (4.2) areequivalent. (cid:3) ROJECTIONS AND ANGLE SUMS OF PERMUTOHEDRA AND OTHER POLYTOPES 15
Normal fans of permutohedra.
Before we start with the proofs of Theorems 3.5 and 3.6, we needto prove a short lemma concerning the interior of a polytope.
Lemma 4.1.
Let P ⊂ R n be a polytope with non-empty interior (i.e. dim P = n ) such that P isgiven by the following affine inequalities P = { x ∈ R n : l ( x ) ≤ , . . . , l m ( x ) ≤ } for some m ∈ N and affine-linear functions l i ( x ) = h x, y i i + b i , where y i ∈ R n \{ } and b i ∈ R , i = 1 , . . . , m . Then, we have int P = { x ∈ R n : l ( x ) < , . . . , l m ( x ) < } . Proof.
Suppose x ∈ R n satisfies the conditions l ( x ) < , . . . , l m ( x ) <
0. Since the functions l , . . . , l m are continuous, we also have l ( y ) < , . . . , l m ( y ) < y in some small enoughneighborhood of x . Thus, x lies in int P .Now let x ∈ P satisfy l i ( x ) = 0 for some i ∈ { , . . . , m } . Then, in each neighborhood of x , wecan find a point y with l i ( y ) >
0. This means that x / ∈ int P , thus completing the proof. (cid:3) Remark . For a j -face F of a polyhedral set P ⊂ R n , the normal cone N F ( P ) is ( n − j )-dimensional. In order to prove this, assume that the linear hull M := lin N F ( P ) is k -dimensionalfor some k < n − j . Then, we have M ⊥ = M ◦ ⊂ N F ( P ) ◦ = T F ( P ). But since dim M ⊥ = n − k > j ,this is a contradiction to the fact that the maximal linear subspace L contained in T F ( P ) is j -dimensional. Also, since L ⊂ T F ( P ), we have L ⊥ ⊃ N F ( P ) and therefore dim N F ( P ) ≤ n − j .Now, let x > . . . > x n be given. For Theorem 3.5, we want to prove that N ( P An ( x , . . . , x n )) = F ( A ( A n − )). Proof of Theorem 3.5.
From Proposition 3.3 we know that each j -face of P An , for a j ∈ { , . . . , n − } ,is uniquely defined by an ordered partition B = ( B , . . . , B n − j ) ∈ R n,n − j of the set { , . . . , n } andgiven by F B = (cid:26) ( t , . . . , t n ) ∈ P An : X i ∈ B ∪ ... ∪ B l t i = x + . . . + x | B ∪ ... ∪ B l | for all l = 1 , . . . , n − j (cid:27) . Now, take a point t ∈ relint F B . We claim that x satisfies the following conditions: X i ∈ B ∪ ... ∪ B l t i = x + . . . + x | B ∪ ... ∪ B l | ∀ l = 1 , . . . , n − j, (4.10)and X i ∈ M t i < x + . . . + x | M | ∀ M ⊂ { , . . . , n } : M / ∈ { B , B ∪ B , . . . , B ∪ . . . ∪ B n − j } . (4.11)In order to prove this, consider the affine subspace L B := (cid:26) ( t , . . . , t n ) ∈ R n : X i ∈ B ∪ ... ∪ B l t i = x + . . . + x | B ∪ ... ∪ B l | for all l = 1 , . . . , n − j (cid:27) , which is of dimension j since the conditions are linearly independent. Then, following Lemma 3.1,we can represent F B as the set of points ( t , . . . , t n ) ∈ L B such that X i ∈ M t i ≤ x + . . . + x | M | ∀ M ⊂ { , . . . , n } : M / ∈ { B , B ∪ B , . . . , B ∪ . . . ∪ B n − j } . Since dim F B = j , the characterization of relint F B in (4.11) follows from Lemma 4.1 applied to theambient affine subspace L B instead of R n .Now, we want to determine the tangent cone T F B ( P An ). By definition, the tangent cone is givenby T F B ( P An ) = { v ∈ R n : t + εv ∈ P An for some ε > } , where t ∈ relint F B . Following Lemma 3.1, for a v ∈ R n , the condition t + εv ∈ P An holds for some ε > n X i =1 ( t i + εv i ) = x + . . . + x n and X i ∈ M ( t i + εv i ) ≤ x + . . . + x | M | ∀ M ⊂ { , . . . , n } . Since t + . . . + t n = x + . . . + x n , the first condition is satisfied if and only if v + . . . + v n = 0.We observe that if we choose ε > M ⊂ { , . . . , n } such that M / ∈ { B , B ∪ B , . . . , B ∪ . . . ∪ B n − j } , due to (4.11). For the sets B , B ∪ B , . . . , B ∪ . . . ∪ B n − j , we obtain that X i ∈ B ∪ ... ∪ B l v i ≤ , following (4.10). Therefore, the tangent cone is given by T F B ( P An ) = (cid:26) v ∈ R n : v + . . . + v n = 0 , X i ∈ B ∪ ... ∪ B l v i ≤ ∀ l = 1 , . . . , n − j − (cid:27) . Thus, the corresponding normal cone is given by N F B ( P An ) = T F B ( P An ) ◦ = { x ∈ R n : ∀ ≤ l ≤ l ≤ n − j ∀ i ∈ B l , i ∈ B l , we have x i ≥ x i } . Note that the conditions of N F B ( P An ) imply x i = x i for all i , i ∈ B l , l = 1 , . . . , n − j . Thecone N F B ( P An ) is an ( n − j )-dimensional cone in the fan F ( A ( A n − )) and it is easy to check that,going through all ordered partitions B ∈ R n,n − j , we obtain all ( n − j )-dimensional cones of the fan N ( A ( A n − )); see, e.g., [15, Section 2.7]. This completes the proof. (cid:3) Now, let x > . . . > x n > N ( P Bn ( x , . . . , x n )) coincides with the fan F ( A ( B n )) generated by the hyperplane arrangement A ( B n ). Proof of Theorem 3.6.
From Proposition 3.4 we know that each j -face of P Bn , for a j ∈ { , . . . , n } ,is uniquely defined by a pair ( B , η ) ∈ T n,n − j , where B = ( B , . . . , B n − j +1 ), and given by F B ,η = (cid:26) ( t , . . . , t n ) ∈ P Bn : X i ∈ B ∪ ... ∪ B l η i t i = x + . . . + x | B ∪ ... ∪ B l | ∀ l = 1 , . . . , n − j (cid:27) . Now, we claim that T F B ,η ( P Bn ) = (cid:26) v ∈ R n : X i ∈ B ∪ ... ∪ B l η i v i ≤ ∀ l = 1 , . . . , n − j (cid:27) . (4.12)In order to prove this, take a point t ∈ relint F B ,η . Then, X i ∈ B ∪ ... ∪ B l η i t i = x + . . . + x | B ∪ ... ∪ B l | ∀ l = 1 , . . . , n − j (4.13) ROJECTIONS AND ANGLE SUMS OF PERMUTOHEDRA AND OTHER POLYTOPES 17 and X i ∈ M | t i | < x + . . . + x | M | ∀ M ⊂ { , . . . , n } : M / ∈ { B , B ∪ B , . . . , B ∪ . . . ∪ B n − j } . (4.14)This can easily be justified in the same way as in the A -case using Lemmas 3.2 and 4.1. Notethat (4.13) implies that sgn t i = η i for all i ∈ B ∪ . . . ∪ B n − j such that t i = 0. Otherwise, if η i = − sgn t i for some i ∈ { , . . . , n } with t i = 0, we would have X B ∪ ... ∪ B l | t i | > X B ∪ ... ∪ B l η i t i = x + . . . + x | B ∪ ... ∪ B l | for some l ∈ { , . . . , n − j } in contradiction to t ∈ P Bn .Now, recall that the tangent cone is defined by T F B ,η ( P Bn ) = { v ∈ R n : t + εv ∈ P Bn for some ε > } for t ∈ relint F B ,η . In view of the characterization of points in P Bn stated in Lemma 3.2, it followsthat v ∈ T F B ,η ( P Bn ) if and only if there exists an ε > X i ∈ M | t i + εv i | ≤ x + . . . + x | M | ∀ M ⊂ { , . . . , n } . For all M ⊂ { , . . . , n } with M / ∈ { B , B ∪ B , . . . , B ∪ . . . ∪ B n − j } this condition is satisfieddue to (4.14) provided ε > t i = 0 for all i ∈ B ∪ . . . ∪ B n − j , the remainingconditions are equivalent to X i ∈ B ∪ ... ∪ B l η i ( t i + εv i ) ≤ x + . . . + x | B ∪ ... ∪ B l | ∀ l = 1 , . . . , n − j. (4.15)This follows from the fact that sgn( t i + εv i ) = sgn t i = η i for ε > X i ∈ B ∪ ... ∪ B l η i v i ≤ , for all l = 1 , . . . , n − j . This proves (4.12). At this point, it remains to prove that t i = 0 for all i ∈ B ∪ . . . ∪ B n − j . In order to do this, assume t i = 0 for some i ∈ B l and some l ∈ { , . . . , n − j } .Defining D i := ( B ∪ . . . ∪ B l ) \{ i } , we have X j ∈ D i η j t j = X j ∈ B ∪ ... ∪ B l η j t j = x + . . . + x | B ∪ ... ∪ B l | , due to (4.13). If D i = B ∪ . . . ∪ B m for some m < l , we obtain x + . . . + x | B ∪ ... ∪ B m | = X j ∈ D i η j t j = x + . . . + x | B ∪ ... ∪ B l | , in contradiction to x i > i = 1 , . . . , n . If, on the other hand, D i = B ∪ . . . ∪ B m for all m < l , we have x + . . . + x | B ∪ ... ∪ B l | = X j ∈ D i η j t j < x + . . . + x | D i | , following (4.14). This is a contradiction to D i ⊂ B ∪ . . . ∪ B l proving that t i = 0 for all i ∈ B ∪ . . . ∪ B n − j . Thus, the normal cone of P Bn at F B ,η is given by N F B ,η ( P Bn ) = T F B ,η ( P Bn ) ◦ = (cid:8) x ∈ R n : ∀ ≤ l ≤ l ≤ n − j ∀ i ∈ B l , i ∈ B l , we have η i x i ≥ η i x i ≥ ∀ i ∈ B n − j +1 we have x i = 0 (cid:9) . The cone N F B ( P Bn ) is an ( n − j )-face of a Weyl chamber of type B n and we can observe that, goingthrough all pairs ( B , η ) ∈ T n,n − j , we obtain all ( n − j )-dimensional cones of the fan N ( A ( B n )); see,e.g., [15, Section 2.4]. This completes the proof. (cid:3) Permutohedra and zonotopes.
In order to prove Proposition 3.13, we need to verify that for x >. . . > x n , the permutohedron P An ( x , . . . , x n ) of type A (respectively, for x > . . . > x n >
0, thepermutohedron P Bn ( x , . . . , x n ) of type B ) is a zonotope if and only if x , . . . , x n are in arithmeticprogression, that is, x j +1 − x j = x j − x j − for all admissible j . Proof of Proposition 3.13.
We will prove both the A - and the B -case together and assume that x > . . . > x n and x > . . . > x n >
0, respectively. In the book of Ziegler [29, Example 7.15], it isshown that P An ( n, n − , . . . ,
1) is a zonotope. By shifting and rescaling, we obtain that P An (cid:0) a + ( n − b, a + ( n − b, . . . , a + b, a (cid:1) is also a zonotope for each a ∈ R and b >
0. Similarly, we can also prove that P Bn ( n, n − , . . . ,
1) isa zonotope and therefore also P An ( a + ( n − b, a + ( n − b, . . . , a + b, a ) for each a > b > P Bn ( n, n − , . . . ,
1) as the following Minkowski sum of linesegments: P Bn ( n, n − , . . . ,
1) = X ≤ i 7→ h c, v i , R n → R , for a vector c ∈ R n , provided the maximizer is unique. Applying a signed permutation, we may assume that c ≥ c ≥ . . . ≥ c n ≥ 0. On the line segment [ − e i − e j , e i − e j ], the function v 7→ h c, v i is uniquelymaximized by the right-hand boundary e i − e j provided c i > c j . For c i = c j , the maximizer is notunique. Therefore, we may assume that c > c > . . . > c n > 0. Then, the unique maximizer of v 7→ h c, v i is given by the sum of the right-hand boundaries of the line segments: v = X ≤ i To prove the other direction, assume that P An ( x , . . . , x n ) with n ≥ P is a zonotope if and only if every 2-dimensional face of P is centrally symmetric [29,p. 200]. Following Proposition 3.3, we know that the convex hull F of the six points( x σ (1) , x σ (2) , x σ (3) , x , x , . . . , x n ) , σ ∈ Sym(3)is a 2-face of P An ( x , . . . , x n ). This face is centrally symmetric around some a = ( a , . . . , a n ). Thismeans that for each vertex z of F , also 2 a − z is a vertex of F . Thus, we obtain the conditions2 a − x , a − x , a − x ∈ { x , x , x } . From x > x > x , we obtain 2 a − x = x and 2 a − x = x and therefore also x + x = 2 x . This yields x − x = x − x . Analogously, by considering moregeneral 2-faces of P An ( x , . . . , x n ), one proves that x j +1 − x j = x j − x j − for all admissible j . Thus, x , . . . , x n are in arithmetic progression.The proof that x , . . . , x n are in arithmetic progression if P Bn ( x , . . . , x n ) is a zonotope followsin the same way as in the A -case since the considered 2-faces of P An are also 2-faces of P Bn , followingProposition 3.4. (cid:3) General position: Proofs of Theorem 3.11 and Corollaries 3.7 and 3.8. In thissection, we prove the equivalences of the general position assumptions stated in Sections 3.2 and 3.3.In fact, Corollaries 3.7 and 3.8 follow from Theorem 3.11.Let P ⊂ R n be polytope such that the normal fan N ( P ) coincides with the fan of a hyperplanearrangement A . For a 1 ≤ d ≤ dim P and a matrix G ∈ R d × n with rank G = d , we want to provethat the following two general position assumptions are equivalent:(G1) The ( n − d )-dimensional linear subspace ker G is in general position with respect to P .(G2) The d -dimensional linear subspace (ker G ) ⊥ is in general position with respect to thehyperplane arrangement A . Proof of Theorem 3.11. Let F ∈ F k ( P ) be a k -face of P for some k ∈ { , . . . , dim P } and let L bethe linear subspace parallel to aff F with the same dimension as aff F , that is, aff F = t + L for some t ∈ R n . Then, the normal cone N F ( P ) is ( n − k )-dimensional, due to Remark 4.2, and coincideswith some ( n − k )-dimensional cone from the fan of A , that is, an ( n − k )-face of the conical mosaicgenerated by A . Thus, lin N F ( P ) can be represented as an intersection of hyperplanes from A andtherefore is an element of the lattice L ( A ).On the other hand, by definition, T F ( P ) contains the linear subspace L . Thus, we have(aff F ) ⊥ = L ⊥ ⊃ T F ( P ) ◦ = N F ( P ). Since both N F ( P ) and (aff F ) ⊥ are ( n − k )-dimensional, weobtain L ⊥ = (aff F ) ⊥ = lin N F ( P ) ∈ L n − k ( A ) . The same argumentation applied backwards shows that, conversely, each ( n − k )-dimensional sub-space K ∈ L ( A ) coincides with lin N F ( P ) for some k -face F of P . If we write aff F = t + L forsome t ∈ R n , as above, then we obtain K = (aff F ) ⊥ = L ⊥ .The equivalence of (G1) and (G2) follows easily from these observations. Condition (G1) isnot satisfied, if and only if dim( L ∩ ker G ) = max { k − d, } for some k ∈ { , . . . , dim P } and some k -dimensional linear subspace L such that aff F = t + L forsome k -face F of P and t ∈ R n . Following the above observation, L ⊥ ∈ L ( A ) anddim (cid:0) (ker G ) ⊥ ∩ L ⊥ (cid:1) = n − dim( L + ker G )= n − (cid:0) dim(ker G ) + dim L − dim( L ∩ ker G ) (cid:1) = d − k + dim( L ∩ ker G )holds true, and we arrive at dim (cid:0) (ker G ) ⊥ ∩ L ⊥ (cid:1) = max { , d − k } . Thus, (ker G ) ⊥ is not in general position with respect to A and therefore, (G2) is not satisfied. Sinceevery K ∈ L ( A ) can be represented as L ⊥ as above, the same argument applies backwards. (cid:3) Now, let x > . . . > x n and G ∈ R d × n be a matrix with rank G = d , 1 ≤ d ≤ n − P An .We want to prove that the following conditions are equivalent:(A1) The ( n − d )-dimensional linear subspace ker G is in general position to P An ( x , . . . , x n ).(A2) The d -dimensional linear subspace (ker G ) ⊥ is in general position with respect to thereflection arrangement A ( A n − ),where A ( A n − ) is the hyperplane arrangement as defined in (3.2). Proof of Corollary 3.7. Following Theorem 3.5, the normal fan N ( P An ) coincides with F ( A ( A n − )).Thus, the equivalence of (A1) and (A2) is a special case of Theorem 3.11. (cid:3) Now, let x > . . . > x n > G ∈ R d × n be a matrix with rank G = d , 1 ≤ d ≤ n = dim P Bn .For Corollary 3.8, we want to verify that the following conditions are equivalent:(B1) The ( n − d )-dimensional linear subspace ker G is in general position to P Bn ( x , . . . , x n ).(B2) The d -dimensional linear subspace (ker G ) ⊥ is in general position with respect to thereflection arrangement A ( B n ),where A ( B n ) is the hyperplane arrangement as defined in (3.3). Proof of Corollary 3.8. Following Theorem 3.6, the normal fan N ( P Bn ) is given by the fan F ( A ( B n )).Thus, the equivalence of (B1) and (B2) is a special case of Theorem 3.11. (cid:3) Face numbers: Proofs of Theorems 3.9, 3.10 and 3.12. In this section, we are going toprove our main results from Section 3.3 on the number of j -faces of projected permutohedra andmore general polytopes. We start with the proof of Theorem 3.12 and proceed with the similarproofs of Theorems 3.9 and 3.10.The following lemma, known as Farkas’ Lemma, will be used in the proof of the namedtheorems. For the proof, we refer to [2, Lemma 2.4] and [13, Lemma 2.1]. Lemma 4.3 (Farkas) . Let C ⊂ R n be a full-dimensional cone and L ⊂ R n a linear subspace. Then,we have int( C ) ∩ L = ∅ ⇔ C ◦ ∩ L ⊥ = { } . Furthermore, we need a formula for the number of regions generated by a hyperplane ar-rangement that are intersected by a linear subspace non-trivially. For the proof, we refer to [10,Theorem 3.1] or [14, Theorem 3.3] in combination with [14, Lemma 3.5]. ROJECTIONS AND ANGLE SUMS OF PERMUTOHEDRA AND OTHER POLYTOPES 21 Lemma 4.4. Let L d ⊂ R n be a d -dimensional linear subspace that is in general position withrespect to a hyperplane arrangement A . Then, the number of regions in R ( A ) (which is the set ofclosed polyhedral cones of the conical mosaic generated by A ) intersected by L d is given by { R ∈ R ( A ) : int R ∩ L d = ∅} = { R ∈ R ( A ) : R ∩ L d = { }} = 2( a n − d +1 + a n − d +3 + . . . ) , where the a k ’s are defined by the characteristic polynomial χ A ( t ) = P nk =0 ( − n − k a k t k . Now, let P ⊂ R n be a polytope such that the normal fan N ( P ) coincides with the fan F ( A ) ofa hyperplane arrangement A . Moreover, let G ∈ R d × n be a matrix with rank G = d ≤ dim P suchthat one of the equivalent general position assumptions (G1) or (G2) is satisfied. For Theorem 3.12,we want to prove that the number of j -faces of the projected polytope GP is given by f j ( GP ) = 2 X M ∈L n − j ( A ) ( a Mn − d +1 + a Mn − d +3 + . . . )for 0 ≤ j < d ≤ dim P , where the numbers a Mk are (up to a sign) the coefficients of the characteristicpolynomial of the hyperplane arrangement A| M := { H ∩ M : H ∈ A , M * H } in the ambientspace M : χ A | M ( t ) = n − j X k =0 ( − n − j − k a Mk t k . Proof of Theorem 3.12. Consider first the case when P is full-dimensional. Let F be a j -face of P and 0 ≤ j < d ≤ n be given. Then, following [1] or [9, Proposition 5.3], GF is a j -face of GP ifand only if int T F ( P ) ∩ ker G = ∅ since the general position assumption (G1) is satisfied. Using Farkas’ Lemma 4.3, this is equivalentto (ker G ) ⊥ ∩ N F ( P ) = { } . Thus, using that N ( P ) = F ( A ) and, in particular, { N F ( P ) : F ∈ F j ( P ) } = F n − j ( A ), we obtain f j ( GP ) = X F ∈F j ( P ) { GF ∈F j ( GP ) } = X F ∈F j ( P ) { (ker G ) ⊥ ∩ N F ( P ) = { }} = X D ∈F n − j ( A ) { (ker G ) ⊥ ∩ D = { }} (4.16)= X M ∈L n − j ( A ) X D ∈F n − j ( A ): D ⊂ M { ((ker G ) ⊥ ∩ M ) ∩ D = { }} , since each ( n − j )-dimensional cone D from the fan F ( A ) of the hyperplane arrangement A iscontained in a unique ( n − j )-dimensional subspace M ∈ L ( A ) that can be represented as anintersection of hyperplanes from A . The ( n − j )-dimensional cones D ∈ F n − j ( A ) with D ⊂ M are the closures of the ( n − j )-dimensional regions generated by the induced arrangement A| M = { H ∩ M : H ∈ A , M * H } in M and therefore, we obtain f j ( GP ) = X M ∈L n − j ( A ) X R ∈R ( A| M ) { ((ker G ) ⊥ ∩ M ) ∩ R = { }} . Due to general position assumption (G2), the subspace (ker G ) ⊥ ∩ M is of codimension n − d in M and additionally in general position with respect to A| M in M . Thus, we can apply Lemma 4.4 tothe ambient linear subspace M and arrive at f j ( GP ) = 2 X M ∈L n − j ( A ) ( a Mn − d +1 + a Mn − d +3 + . . . ) , (4.17)which completes the proof in the full-dimensional case.Now, suppose p := dim P < n . We want to restrict all arguments to the p -dimensionallinear subspace L satisfying aff P = t + L for some t ∈ R n , and then apply the already knownfull-dimensional case in the ambient space L . At first, we observe that rank( G | L ) = d , sincedim ker( G | L ) = dim( L ∩ ker G ) = p − d ≥ G is in general position with respect to P dueto general position assumption (G1). Furthermore, we need to verify whether the conditions (G1)and (G2) also hold in the restricted case where n is replaced by p , G is replaced by the restriction G | L of G to L , and A is replaced by A| L = { H ∩ L : H ∈ A , L * H } = { H ∩ L : H ∈ A} . Thelast equation is due to L ⊥ ⊂ lin N F ( P ) for all faces F of P , and therefore, L ⊥ ⊂ H for all H ∈ A ,since the linear hull lin N F ( P ) coincides with an intersection of hyperplanes from A . Thus, wealso observe that the elements of A| L and A are in one-to-one correspondence via the mapping H ′ H ′ + L ⊥ and, the inverse map is given by H ∩ L ← [ H .Also, following (G1) for P in R n , ker( G | L ) is in general position with respect to K , for eachlinear subspace K such that aff F = t + K for some face F of P , sincedim( K ∩ ker( G | L )) = dim( K ∩ L ∩ ker G ) = dim( K ∩ ker G ) . Thus, (G1) is also satisfied if we restrict all objects to L . Then, (G2) is also satisfied in the restrictedversion due to the equivalence of (G1) and (G2) proved in Theorem 3.11. Thus, we can apply (4.17)in the restricted case to obtain f j ( GP ) = 2 X M ′ ∈L p − j ( A| L ) (cid:0) a M ′ p − d +1 + a M ′ p − d +3 + . . . (cid:1) , since ( A| L ) | M ′ = A| M ′ and therefore χ ( A| L ) | M ′ ( t ) = χ A| M ′ ( t ). Next we observe that the linearsubspaces M ′ ∈ L p − j ( A| L ) are in one-to-one correspondence to the linear subspaces M ∈ L n − j ( A )via M ′ M ′ + L ⊥ =: M . Following the Whitney formula for the characteristic polynomial (3.8)and the identity A| ( M ′ + L ⊥ ) = ( A| M ′ ) + L ⊥ , we obtain the relation χ A| ( M ′ + L ⊥ ) ( t ) = χ ( A| M ′ )+ L ⊥ ( t ) = t n − p χ A| M ′ ( t ) , for all M ′ ∈ L p − j ( A| L ), and thus, a M ′ k = a M ′ + L ⊥ k + n − p . Hence, we arrive at f j ( GP ) = 2 X M ′ ∈L p − j ( A| L ) (cid:0) a M ′ + L ⊥ n − d +1 + a M ′ + L ⊥ n − d +3 + . . . (cid:1) = 2 X M ∈L n − j ( A ) (cid:0) a Mn − d +1 + a Mn − d +3 + . . . (cid:1) , which completes the proof. (cid:3) ROJECTIONS AND ANGLE SUMS OF PERMUTOHEDRA AND OTHER POLYTOPES 23 Permutohedron of type A . Now, we are going to prove the formulas for the number of faces of theprojected permutohedra of types A and B . Before starting with the proof of Theorem 3.9, we needto introduce the Weyl chambers of type A n − and the corresponding reflection group and reflectionarrangement. The reflection group G ( A n − ) of type A n − acts on R n by permuting the coordinatesin an arbitrary way, that is, the n ! elements of G ( A n − ) are the linear mappings g σ : R n → R n , ( β , . . . , β n ) ( β σ (1) , . . . , β σ ( n ) ) , where σ ∈ Sym( n ). The closed fundamental Weyl chamber of type A n − is the cone C ( A n − ) = { ( β , . . . , β n ) ∈ R n : β ≥ . . . ≥ β n } . Then, the closed Weyl chambers of type A n − are the cones of the form g C ( A n − ), where g ∈G ( A n − ), that is, the cones C Aσ := { ( β , . . . , β n ) ∈ R n : β σ (1) ≥ β σ (2) ≥ . . . ≥ β σ ( n ) } , σ ∈ Sym( n ) . Equivalently, the Weyl chambers can be defined as the conical mosaic generated by the reflectionarrangement A ( A n − ) consisting of the hyperplanes { β ∈ R n : β i = β j } , ≤ i < j ≤ n. Note that with this notation, the permutohedron P An ( x , . . . , x n ) for a point ( x , . . . , x n ) ∈ R n isjust the convex hull of all points g ( x , . . . , x n ), where g ∈ G ( A n − ).In order to prove Theorem 3.9, we need to evaluate the number of j -faces F ∈ F j ( A ( A n − ))that are intersected non-trivially by a d -dimensional linear subspace satisfying some general positionassumption. Lemma 4.5. The number of j -faces of Weyl chambers of type A n − (where each face is countedexactly once) intersected non-trivially by a d -dimensional subspace L d in general position to thereflection arrangement A ( A n − ) is given by X F ∈F j ( A ( A n − )) { F ∩ L d = { }} = 2 (cid:26) nj (cid:27)(cid:18)(cid:20) jn − d + 1 (cid:21) + (cid:20) jn − d + 3 (cid:21) + . . . (cid:19) , for all j ∈ { , . . . , n } . Recall that the numbers (cid:2) nk (cid:3) and (cid:8) nk (cid:9) denote the Stirling numbers of the first and second kind,respectively, as defined in Section 3.3.Note that for j = n , this result is already known. That is, the number of Weyl chambers oftype A n − intersected by a d -dimensional subspace L d in general position with respect to A ( A n − )is given by X F ∈F n ( A ( A n − )) { F ∩ L d = { }} = X σ ∈ Sym( n ) { C Aσ ∩ L d = { }} = 2 (cid:18)(cid:20) nn − d + 1 (cid:21) + (cid:20) nn − d + 3 (cid:21) + . . . (cid:19) ;(4.18)see [14, Theorem 3.4]. The proof of Lemma 4.5 is similar to that of [15, Theorem 2.8], where arelated formula has been established in a setting where the faces are counted with certain non-trivialmultiplicities. Proof of Lemma 4.5. The j -dimensional faces of the Weyl chamber C Aσ are enumerated by collec-tions of indices 1 ≤ i < . . . < i j − ≤ n − C Aσ ( i , . . . , i j − ) := { β ∈ R n : β σ (1) = . . . = β σ ( i ) ≥ . . . ≥ β σ ( i j − +1) = . . . = β σ ( n ) } It is easy to see that C Aσ ( i , . . . , i j − ) may also be a j -face of another Weyl chamber C Aσ ′ for some σ ′ = σ (the permutations inside each group of equations of the defining conditions of C Aσ ( i , . . . , i j − )can be chosen arbitrary without changing the face itself).Now, we will introduce a notation for all j -faces of all Weyl chambers of type A n − in the waythat each face is counted exactly once. Recall that R n,j denotes the set of all ordered partitions B = ( B , . . . , B j ) of the set { , . . . , n } into j disjoint non-empty subsets. For such an orderedpartition B , we define the polyhedral cone Q B = (cid:8) β ∈ R n : for all 1 ≤ l ≤ l ≤ j and i ∈ B l , i ∈ B l we have β i ≥ β i (cid:9) . Note that these conditions imply β i = β i for all i , i ∈ B l and all 1 ≤ l ≤ j . We can observe thateach j -face F ∈ F j ( A ( A n − )) coincides with Q B for a unique ordered partition B ∈ R n,j and that,conversely, each cone Q B is a j -face from F j ( A ( A n − )). Thus, we have X F ∈F j ( A ( A n − )) { F ∩ L d = { }} = X B∈R n,j { Q B ∩ L d = { }} . Now, we want to evaluate the right hand-side of the above equation. At first, consider thecase j ≤ n − d . Since L d is in general position with respect to A ( A n − ), we know that for each Q B dim( L d ∩ lin Q B ) = max( j + d − n, 0) = 0 , since lin Q B ∈ L ( A n − ). Therefore, Lemma 4.5 becomes trivial since both sides vanish. From nowon, assume that j ≥ n − d + 1. The j -dimensional linear hull of Q B is given by W B = (cid:8) β ∈ R n : for all 1 ≤ l ≤ j and i , i ∈ B l we have β i = β i (cid:9) . Using y l := β i , where i ∈ B l is arbitrary and l = 1 , . . . , j , as coordinates on W B , allows us toidentify this linear hull with R j . This identification is linear (which is sufficient for what follows)but not isometric. The subspace W B naturally decomposes into j ! Weyl chambers of type A j − ofthe form W B ( π ) = { ( β , . . . , β n ) ∈ W B : y π (1) ≥ . . . ≥ y π ( j ) } , where π ∈ Sym( j ). Note that the Weyl chamber in W B corresponding to the identity permutation π ( i ) = i for all 1 ≤ i ≤ j is Q B itself. Since L d ⊂ R n has dimension d and is in general withrespect to the reflection arrangement A ( A n − ) and since W B is an intersection of hyperplanes from A ( A n − ), as mentioned above, it follows that the subspace L d ∩ W B ⊂ W B has dimension d − n + j and is in general position with respect to the reflection arrangement of type A j − in W B . This canbe easily verified using the definition (3.4). Thus, we obtain X π ∈ Sym( j ) { L d ∩ W B ( π ) = { }} = X π ∈ Sym( j ) { ( L d ∩ W B ) ∩ W B ( π ) = { }} = 2 (cid:18)(cid:20) jn − d + 1 (cid:21) + (cid:20) jn − d + 3 (cid:21) + . . . (cid:19) , following (4.18) applied to L d ∩ W B in the ambient linear subspace W B . If we take the sum overall ordered partitions B ∈ R n,j , we arrive at X B∈R n,j X π ∈ Sym( j ) { L d ∩ W B ( π ) = { }} = 2 X B∈R n,j (cid:18)(cid:20) jn − d + 1 (cid:21) + (cid:20) jn − d + 3 (cid:21) + . . . (cid:19) . (4.19)The right-hand side of (4.19) can be easily computed using that the number of (unordered) parti-tions of { , . . . , n } into j non-empty subsets is given by the Stirling number (cid:8) nj (cid:9) of the second kind.Therefore, the number of ordered partitions is given by j ! (cid:8) nj (cid:9) , since the sets of each partition canbe arranged in an arbitrary order. Moreover, since the j -face Q B can be represented as W B ′ ( π ′ ) in ROJECTIONS AND ANGLE SUMS OF PERMUTOHEDRA AND OTHER POLYTOPES 25 j ! ways and the sets B , . . . , B j are (up to their order) the same in all representations, the left-handside of (4.19) can be rewritten as j ! X B∈R n,j { Q B ∩ L d = { }} . Combining these results yields X F ∈F j ( A ( A n − )) { F ∩ L d = { }} = X B∈R n,j { Q B ∩ L d = { }} = 2 (cid:26) nj (cid:27)(cid:18)(cid:20) jn − d + 1 (cid:21) + (cid:20) jn − d + 3 (cid:21) + . . . (cid:19) , which completes the proof. (cid:3) Now, we turn to the proof of Theorem 3.9. Let G ∈ R d × n be a matrix with rank G = d satisfying one of the equivalent general position assumptions (A1) or (A2). Assume x > . . . > x n .We want to show that f j ( G P An ) = 2 (cid:26) nn − j (cid:27) (cid:18)(cid:20) n − jn − d + 1 (cid:21) + (cid:20) n − jn − d + 3 (cid:21) + . . . (cid:19) holds for all 0 ≤ j < d ≤ n − Proof of Theorem 3.9. Since P An is a polytope whose normal fan coincides with the fan generatedby A ( A n − ) and the equivalent general positions assumptions (A1) and (A2) are satisfied, we canapply (4.16) from the proof of Theorem 3.12 with P replaced by P An and A replaced by A ( A n − ),and obtain f j ( G P An ) = X D ∈F n − j ( A ( A n − )) { (ker G ) ⊥ ∩ D = { }} . As was shown in the proof of Theorem 3.12, (4.16) was applicable although P An is not full-dimensional. Following Lemma 4.5, we arrive at f j ( G P An ) = 2 (cid:26) nn − j (cid:27)(cid:18)(cid:20) n − jn − d + 1 (cid:21) + (cid:20) n − jn − d + 3 (cid:21) + . . . (cid:19) , since (ker G ) ⊥ has dimension d and is in general position with respect to the reflection arrangement A ( A n − ), due to (A1). This completes the proof. (cid:3) Permutohedron of type B . For the proof of Theorem 3.10, we need to introduce the Weyl chambersof type B n . The reflection group G ( B n ) acts on R n by permuting the coordinates in an arbitraryway and multiplying an arbitrary number of coordinates by − 1. Thus, the 2 n n ! elements of G ( B n )are the linear mappings g ε,σ : R n → R n , ( β , . . . , β n ) ( ε β σ (1) , . . . , ε n β σ ( n ) ) , where σ ∈ Sym( n ) and ε = ( ε , . . . , ε n ) ∈ {± } n . The closed fundamental Weyl chamber of type B n is the cone C ( B n ) = { ( β , . . . , β n ) ∈ R n : β ≥ . . . ≥ β n ≥ } . Then, the closed Weyl chambers of type B n are the cones g C ( A n − ), where g ∈ G ( B n ), that is, thecones C Bε,σ := { ( β , . . . , β n ) ∈ R n : ε β σ (1) ≥ ε β σ (2) ≥ . . . ≥ ε n β σ ( n ) ≥ } , ε ∈ {± } n , σ ∈ Sym( n ) . Equivalently, the Weyl chambers of type B n can be defined as the conical mosaic generated by thereflection arrangement A ( B n ) consisting of the hyperplanes (3.3). Again, we observe that with this notation, the permutohedron P Bn ( x , . . . , x n ) for a point ( x , . . . , x n ) ∈ R n is just the convex hullsof all points g ( x , . . . , x n ), where g ∈ G ( B n ).In order to show Theorem 3.10, we need to prove the following lemma. Lemma 4.6. The number of j -faces of Weyl chambers of type B n (where each face is countedexactly once) intersected non-trivially by a d -dimensional subspace L d in general position to A ( B n ) is given by X F ∈F j ( A ( B n )) { F ∩ L d = { }} = 2 T ( n, j ) (cid:0) B ( j, n − d + 1) + B ( j, n − d + 3) + . . . (cid:1) , for all j ∈ { , . . . , n } . Recall that the numbers B ( n, k ) and T ( n, k ) are the B -analogues to the Stirling numbers ofthe first and second kind, respectively, as defined in (3.6) and (3.7). Also note that for j = n ,this lemma is already known. That is, the number of Weyl chambers of type B n intersected by a d -dimensional subspace L d in general position with respect to A ( B n ) is given by X F ∈F n ( A ( B n )) { F ∩ L d = { }} = X ( ε,σ ) ∈{± } n × Sym( n ) { C Bε,σ ∩ L d = { }} = 2 (cid:0) B ( n, n − d + 1) + B ( n, n − d + 3) + . . . (cid:1) ; (4.20)see [14, Theorem 3.4] or [15, Theorem 2.4]. The proof of Lemma 4.6 is similar to that of [15,Theorem 2.1]. Proof of Lemma 4.6. Since this is proven similarly to Lemma 4.5, we will not give the proof infull detail. At first, we will introduce the notation for all j -faces of all Weyl chambers of type B n in the way that each face is counted exactly once. Recall that T n,j denotes the set of all pairs( B , η ), where B = ( B , . . . , B j +1 ) is an ordered partition of the set { , . . . , n } into j + 1 disjointdistinguishable subsets such that B , . . . , B j are non-empty, whereas B j +1 may be empty or not,and η : B ∪ . . . ∪ B j → {± } . For ease of notation set η ( i ) = η i , i ∈ B ∪ . . . ∪ B j . The j -face from F j ( A ( B n )) corresponding to ( B , η ) is then given by Q B ,η = (cid:8) β ∈ R n : for all 1 ≤ l ≤ l ≤ j and i ∈ B l , i ∈ B l we have η i β i ≥ η i β i ≥ i ∈ B j +1 we have β i = 0 (cid:9) . Note that these conditions imply η i β i = η i β i for all i , i ∈ B l and all 1 ≤ l ≤ j . Similarly tothe A -case, we obtain X F ∈F j ( A ( B n )) { F ∩ L d = { }} = X ( B ,η ) ∈T n,j { Q B ,η ∩ L d = { }} . Now, we want to evaluate the right hand-side of the above equation. Like in the A -case,Theorem 3.10 becomes trivial for j ≤ n − d . Thus, assume that j ≥ n − d + 1. The j -dimensionallinear hull of Q B ,η is given by W B ,η = (cid:8) β ∈ R n : for all 1 ≤ l ≤ j and i , i ∈ B l we have η i β i = η i β i ;for all i ∈ B j +1 we have β i = 0 (cid:9) . We can use y l := η i β i for i ∈ B l and l = 1 , . . . , j as coordinates on W B . Then, we can identify thislinear hull with R j and naturally decompose W B ,η into 2 j j ! Weyl chambers of type B n of the form W B ,η ( π, δ ) = { ( β , . . . , β n ) ∈ W B ,η : δ y π (1) ≥ . . . ≥ δ j y π ( j ) ≥ } , ROJECTIONS AND ANGLE SUMS OF PERMUTOHEDRA AND OTHER POLYTOPES 27 where π ∈ Sym( j ) and δ = ( δ , . . . , δ j ) ∈ {± } j . Since L d ⊂ R n has dimension d and is in generalwith respect to the reflection arrangement A ( B n ) it follows from the definition that the subspace L d ∩ W B ⊂ W B has dimension d − n + j and is in general position with respect to the reflectionarrangement of type B j in W B ,η . Thus, we obtain X ( π,δ ) ∈ Sym( j ) ×{± } j { L d ∩ W B ,η ( π,δ ) = { }} = 2 (cid:0) B ( j, n − d + 1) + B ( j, n − d + 3) + . . . (cid:1) , following (4.20) applied to L d ∩ W B ,η in the ambient linear subspace W B ,η . We arrive at X ( B ,η ) ∈T n,j X ( π,δ ) ∈ Sym( j ) ×{± } j { L d ∩ W B ,η ( π,δ ) = { }} (4.21)= 2 X ( B ,η ) ∈T n,j ( B ( j, n − d + 1) + B ( j, n − d + 3) + . . . )= 2( B ( j, n − d + 1) + B ( j, n − d + 3) + . . . ) · T n,j . The number of elements in T n,j is given by T n,j = n − j X r =0 n − r j ! (cid:18) nr (cid:19)(cid:26) n − rj (cid:27) = 2 j j ! n X r = j r − j (cid:18) nr (cid:19)(cid:26) rj (cid:27) = 2 j j ! T ( n, j ) . The last equation follows from (3.7) and the first equation can be proved as follows. There are (cid:0) nr (cid:1) possibilities to fix the elements of the set B j +1 , given it has cardinality r ∈ { , . . . , n − j } . Then,there are 2 n − r possibilities for the choice of signs of the other n − r elements in B , . . . , B j . At last,the number of ordered partitions of { , . . . , n }\ B j +1 into j non-empty subsets is given by j ! (cid:8) n − rj (cid:9) .Summing over all admissible values of r yields the first equation.Moreover, since each j -face Q B ,η can be represented as W B ′ ,η ′ ( π ′ , δ ′ ) in 2 j j ! ways, the sumin (4.21) can be rewritten as 2 j j ! X ( B ,η ) ∈T n,j { Q B ,η ∩ L d = { }} . Combining the above yields X F ∈F j ( A ( B n )) { F ∩ L d = { }} = 2 T ( n, j ) (cid:0) B ( j, n − d + 1) + B ( j, n − d + 3) + . . . (cid:1) , which completes the proof. (cid:3) Now, we are finally able to present the proof of Theorem 3.10. Let n ≥ d and G ∈ R d × n be amatrix with rank G = d satisfying one of the equivalent general position assumptions (B1) or (B2).Assume x > . . . > x n > 0. We want to prove that f j ( G P Bn ) = 2 T ( n, n − j ) (cid:0) B ( n − j, n − d + 1) + B ( n − j, n − d + 3) + . . . (cid:1) holds for all 0 ≤ j < d ≤ n . Proof of Theorem 3.10. Since P Bn is a polytope whose normal fan coincides with the fan generatedby A ( B n ) and the equivalent general positions assumptions (B1) and (B2) are satisfied, we canapply (4.16) with P replaced by P Bn and A replaced by A ( B n ). Thus, we obtain f j ( G P Bn ) = X D ∈F n − j ( A ( B n )) { D ∩ (ker G ) ⊥ = { }} = 2 T ( n, n − j ) (cid:0) B ( n − j, n − d + 1) + B ( n − j, n − d + 3) + . . . (cid:1) . where we applied Lemma 4.6 in the last step. (cid:3) Angle Sums: Proofs of Theorems 3.14, 3.15 and 3.16. The main ingredient in theproofs of Theorems 3.14, 3.15 and 3.16 is the following formula: X F ∈F j ( P ) γ d ( T F ( P )) = f j ( P ) − E f j ( GP ) (4.22)for a polyhedral set P ⊂ R n with non-empty interior, a Gaussian random matrix G ∈ R d × n andall 0 ≤ j < d ≤ n . This formula (with the Gaussian projection G replaced by a projection on a d -dimensional uniform subspace) is often used throughout the existing literature, most prominentlyby Affentranger and Schneider [1], and a detailed proof can be found in [9, Theorem 4.8]. Resultsfrom Baryshnikov and Vitale [5] imply that the f -vector of the Gaussian projection of a polyhedralset P has the same distribution as the f -vector of a uniform projection of P .Suppose x > . . . > x n . For Theorem 3.14, we want to show that the formulas X F ∈F j ( P An ) γ d ( T F ( P An )) = 2 (cid:26) nn − j (cid:27) ∞ X l =0 (cid:20) n − jn − d − l − (cid:21) , X F ∈F j ( P An ) υ d ( T F ( P An )) = (cid:26) nn − j (cid:27)(cid:20) n − jn − d (cid:21) hold for all 1 ≤ j ≤ d ≤ n − Proof of Theorem 3.14. Let first 1 ≤ j < d ≤ n − G ∈ R d × n meaning that its entries are independent and standard Gaussian distributed random variables.This matrix has rank G = d a.s. since the rows of G are a.s. linearly independent; see, e.g., the proofof Theorem 4.17 in [9]. Also, the random matrix G satisfies general position assumption (A1) a.s.This can be justified by noting that ker G has a rotationally invariant distribution which impliesthat ker G is uniformly distributed in the Grassmannian of all ( n − d )-dimensional subspaces of R n . This means that ker G is in general position with respect to each subspace L ⊂ R n a.s.,following [21, Lemma 13.2.1]. In particular, ker G is in general position with respect to P An . Thuswe can apply (4.22) and Theorem 3.9 and obtain X F ∈F j ( P An ) γ d ( T F ( P An )) = f j ( P An ) − E f j ( G P An )= f j ( P An ) − (cid:26) nn − j (cid:27) (cid:18)(cid:20) n − jn − d + 1 (cid:21) + (cid:20) n − jn − d + 3 (cid:21) + . . . (cid:19) = ( n − j )! (cid:26) nn − j (cid:27) − (cid:26) nn − j (cid:27) (cid:18)(cid:20) n − jn − d + 1 (cid:21) + (cid:20) n − jn − d + 3 (cid:21) + . . . (cid:19) , where we used Proposition 3.3 to determine the number of j -faces of P An , which coincides with thenumber of ordered partitions of the set { , . . . , n } into n − j non-empty subsets. Note that for x > . . . > x n the permutohedron P An is only ( n − P An since the Grassmann angles do not depend on thedimension of ambient linear subspace.We can simplify the above formula using that (cid:20) n (cid:21) + (cid:20) n (cid:21) + . . . = (cid:20) n (cid:21) + (cid:20) n (cid:21) + . . . = n !2 . ROJECTIONS AND ANGLE SUMS OF PERMUTOHEDRA AND OTHER POLYTOPES 29 Thus, we obtain X F ∈F j ( P An ) γ d ( T F ( P An )) = 2 (cid:26) nn − j (cid:27) (cid:18) ( n − j )!2 − (cid:18)(cid:20) n − jn − d + 1 (cid:21) + (cid:20) n − jn − d + 3 (cid:21) + . . . (cid:19)(cid:19) = 2 (cid:26) nn − j (cid:27) ∞ X l =0 (cid:20) n − jn − d − l − (cid:21) , which completes the proof of (3.12).To deduce (3.11) we use the relation between the Grassmann angles and the conic intrinsicvolumes (2.2), namely υ d ( C ) = 12 γ d − ( C ) − γ d +1 ( C ) , for all d ∈ { , . . . , n } provided C is not a linear subspace, where in the cases d = 0 and d = n wehave to define γ − ( C ) = 1 and γ n +1 ( C ) = 0. For d ∈ { j + 1 , . . . , n − } , we have X F ∈F j ( P An ) υ d ( T F ( P An )) = 12 X F ∈F j ( P An ) (cid:0) γ d − ( T F ( P An )) − γ d +1 ( T F ( P An )) (cid:1) = (cid:26) nn − j (cid:27)(cid:18) ∞ X l =0 (cid:20) n − jn − d − l (cid:21) − ∞ X l =0 (cid:20) n − jn − d − l − (cid:21)(cid:19) = (cid:26) nn − j (cid:27)(cid:20) n − jn − d (cid:21) . To complete the proof, note that in the case j = d we have υ j ( T F ( P An )) = υ n − j ( N F ( P An )), whichis the solid angle of N F ( P An ). By Theorem 3.5, the sum of these angles over all F ∈ F j ( P An ) is thenumber of linear subspaces in L n − j ( A ( A n − )), which is given by (cid:8) nn − j (cid:9) . (cid:3) Now, suppose x > . . . > x n > 0. In order to prove Theorem 3.15, we need to verify theformulas X F ∈F j ( P Bn ) υ d ( T F ( P Bn )) = T ( n, n − j ) B ( n − j, n − d ) , (4.23) X F ∈F j ( P Bn ) γ d ( T F ( P Bn )) = 2 T ( n, n − j ) ∞ X l =0 B ( n − j, n − d − l − 1) (4.24)for 0 ≤ j ≤ d ≤ n , where the numbers B ( n, k ) and T ( n, k ) are the B -analogues to the Stirlingnumbers of first and second kind, respectively, as defined in (3.6) and (3.7). Proof of Theorem 3.15. Similarly to the proof of Theorem 3.14, we use the formula (4.22). Let G ∈ R d × n be a Gaussian random matrix and let 0 ≤ j < d ≤ n . As seen in the proof ofTheorem 3.14, ker G = d a.s. and ker G is in general position with respect to each linear subspace L ⊂ R n a.s. Thus, ker G is also in general position with respect to P Bn a.s. and therefore, generalposition assumption (B1) is a.s. satisfied. Applying (4.22) and Theorem 3.10, we obtain X F ∈F j ( P Bn ) γ d ( T F ( P Bn )) = f j ( P Bn ) − E f j ( G P Bn )= f j ( P Bn ) − T ( n, n − j ) (cid:0) B ( n − j, n − d + 1) + B ( n − j, n − d + 3) + . . . (cid:1) . By Proposition 3.4, the number of j -faces of P Bn is given by the number of pairs ( B , η ) ∈ T n,n − j .This number was already computed in the proof of Lemma 4.6 and is given by f j ( P Bn ) = T n,n − j = 2 n − j ( n − j )! T ( n, n − j ) . Using the equation B ( n, 1) + B ( n, 3) + . . . = B ( n, 0) + B ( n, 2) + . . . = 2 n − n ! , we obtain X F ∈F j ( P Bn ) γ d ( T F ( P Bn ))= 2 T ( n, n − j ) (cid:0) n − j − ( n − j )! − ( B ( n − j, n − d + 1) + B ( n − j, n − d + 3) + . . . ) (cid:1) = 2 T ( n, n − j ) ∞ X l =0 B ( n − j, n − d − l − d ∈ { j + 1 , . . . , n } X F ∈F j ( P Bn ) υ d ( T F ( P Bn )) = 12 X F ∈F j ( P Bn ) (cid:0) γ d − ( T F ( P Bn )) − γ d +1 ( T F ( P Bn )) (cid:1) = T ( n, n − j ) B ( n − j, n − d ) . The case j = d is treated similarly to the A -case. (cid:3) At last, we want to prove Theorem 3.16. Let P ⊂ R n be a polytope whose normal fan N ( P )coincides with the fan F ( A ) of a hyperplane arrangement A in R n . Then, we want to prove theformulas X F ∈F j ( P ) υ d ( T F ( P )) = X M ∈L n − j ( A ) a Mn − d , X F ∈F j ( P ) γ d ( T F ( P )) = 2 X M ∈L n − j ( A ) ( a Mn − d − + a Mn − d − + . . . )for 0 ≤ j ≤ d ≤ dim P , where the numbers a Mk are defined by χ A | M ( t ) = P jk =0 ( − j − k a Mk t k and L j ( A ) denotes the set of j -dimensional subspaces from the lattice of A . Proof of Theorem 3.16. Let G ∈ R d × n be a Gaussian random matrix. In the same way as inthe proof of Theorem 3.14, we can show that G satisfies the equivalent general position assump-tions (G1) and (G2) a.s. Again, we use the formula (4.22) applied to the ambient affine subspaceaff P . Together with Theorem 3.12, we obtain X F ∈F j ( P ) γ d ( T F ( P )) = f j ( P ) − E f j ( GP )= f j ( P ) − X M ∈L n − j ( A ) ( a Mn − d +1 + a Mn − d +3 + . . . ) (4.25)for all 0 ≤ j < d ≤ dim P . For a j -face F of P , the corresponding normal cone N F ( P ) is ( n − j )-dimensional and since the normal fan of P is given by the fan F ( A ), we obtain f j ( P ) = F n − j ( A ). ROJECTIONS AND ANGLE SUMS OF PERMUTOHEDRA AND OTHER POLYTOPES 31 The number of regions of the arrangements A can be expressed through the characteristicpolynomial by means of the Zaslavsky formula R ( A ) = ( − n χ A ( − n − j )-dimensional cone of the fan F ( A ) is uniquely containedin an ( n − j )-dimensional subspace M ∈ L n − j ( A ), and since the ( n − j )-dimensional cones of F ( A )contained in M are the cones of the conical mosaic in M generated by the induced arrangement A| M := { H ∩ M : H ∈ A , M * H } , we obtain F n − j ( A ) = X M ∈L n − j ( A ) { F ∈ F n − j ( A ) : F ⊂ M } = X M ∈L n − j ( A ) R ( A| M )= X M ∈L n − j ( A ) ( − n − j χ A| M ( − . Furthermore, we have χ A| M ( − 1) = n − j X k =0 ( − n − j − k a Mk ( − k = ( − n − j n − j X k =0 a Mk . Combining these results, we arrive at f j ( P ) = X M ∈L n − j ( A ) n − j X k =0 a Mk . Now, we can insert this formula for f j ( P ) in (4.25) together with the identity a M + a M + . . . = a M + a M + . . . , see [14, Remark 3.2], and obtain X F ∈F j ( P ) γ d ( T F ( P )) = X M ∈L n − j ( A ) n − j X k =0 a Mk − X M ∈L n − j ( A ) ( a Mn − d +1 + a Mn − d +3 + . . . )= 2 X M ∈L n − j ( A ) ( a Mn − d − + a Mn − d − + . . . )for 0 ≤ j < d ≤ dim P .The formula for the sums of the conic intrinsic volumes follows from relation (2.2): X F ∈F j ( P ) υ d ( T F ( P )) = 12 X F ∈F j ( P ) (cid:0) γ d − ( T F ( P )) − γ d +1 ( T F ( P )) (cid:1) = X M ∈L n − j ( A ) a Mn − d for d ∈ { j + 1 , . . . , n } . In the case j = d , the same argument as used at the end of the proof ofTheorem 3.14 shows that P F ∈F j ( P ) υ j ( T F ( P )) = L n − j ( A ) = P M ∈L n − j ( A ) a Mn − j since a Mn − j = 1.The proof is complete. (cid:3) References [1] F. Affentranger and R. Schneider. 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Translated from the Russian by G. H. Lawden.[29] G. M. Ziegler. Lectures on polytopes . Springer-Verlag, New York, 1995. Thomas Godland: Institut f¨ur Mathematische Stochastik, Westf¨alische Wilhelms-Universit¨atM¨unster, Orl´eans-Ring 10, 48149 M¨unster, Germany E-mail address : [email protected] Zakhar Kabluchko: Institut f¨ur Mathematische Stochastik, Westf¨alische Wilhelms-Universit¨atM¨unster, Orl´eans–Ring 10, 48149 M¨unster, Germany E-mail address ::