Random Polyhedral Cones I: Distributional Results via Gale Duality

RANDOM POL YHEDRAL CONES I: DISTRIBUTIONAL RESUL TS VIA GALE DUALITY ZAKHAR KABLUCHK O Abstract. Let U 1 , . . . , U n b e indep enden t random v ectors uniformly distributed on the unit sphere S d − 1 ⊆ R d , where n ≥ d , and consider the random p olyhedral cone W n,d := p os( U 1 , . . . , U n ) = { λ 1 U 1 + . . . + λ n U n : λ 1 ≥ 0 , . . . , λ n ≥ 0 } . W e establish sev eral distributional results for W n,d and the associated spherical p olytop e W n,d ∩ S d − 1 . Our main con tributions include: (i) Let α d denote the solid angle of W d,d and write m ( d, k ) := E [ α k d ] for its k -th momen t. W e pro v e the symmetry m ( d, k ) = m ( k , d ). As an application, w e compute V ar[ α d ] = 2 − d ( d + 1) − 1 − 4 − d and deriv e a closed formula for the third momen t. (ii) F or n = d + 1 , d + 2 , d + 3 w e determine the probability that W n,d ∩ S d − 1 is a spherical simplex, a spherical analogue of the classical Sylvester problem. In the case n = d + 2 w e also determine the distribution of the num ber of vertices of W d +2 ,d ∩ S d − 1 . (iii) Let f ℓ ( W n,d ) denote the num ber of ℓ -dimensional faces of W n,d . W e prov e a distributional limit theorem for f ℓ ( W n,d ) in the regime n = d + k and ℓ = d − q , where k, q ∈ N are ﬁxed and d → ∞ . The limit law is a weigh ted sum of independent chi squared v ariables, with w eigh ts giv en b y explicit eigen v alues of a conv olution op erator on the sphere. (iv) Let I 1 , I 2 ⊆ [ n ] b e such that I 1 ∪ I 2 = [ n ] and max(# I 1 , # I 2 ) ≤ d − 1. W e prov e that the “face ev en ts” { p os( U i : i ∈ I r ) is a face of W n,d } , r = 1 , 2, are indep endent. A unifying ingredient is an explicit coupling pro ducing i.i.d. uniform vectors U 1 , . . . , U n ∈ S d − 1 together with i.i.d. uniform vectors V 1 , . . . , V n ∈ S n − d − 1 whose asso ciated oriented matroids are Gale dual; faces of p os( U 1 , . . . , U n ) corresp ond to complementary subconﬁgurations of V 1 , . . . , V n that p ositiv ely span R n − d . This correspondence yields, in particular, a U -statistic represen tation of f ℓ ( W n,d ). Contents 1. In tro duction 2 1.1. Random p olyhedral cones 2 1.2. Main results 3 1.3. Common technique: Gale duality 4 1.4. Notation 4 2. Linear Gale dualit y for random vector conﬁgurations 5 2.1. Linear Gale dualit y 5 2020 Mathematics Subje ct Classiﬁc ation. Primary: 60D05, 52A22; Secondary: 52A55, 52B11, 52B35, 52B05, 33C55, 60F05. Key wor ds and phr ases. Sto c hastic geometry , geometric probability , random p olyhedral cones, random spheri- cal p olytop es, solid angles, Gale duality , spherical Sylvester problem, U -statistics, F unk–Heck e formula, spherical harmonics. 1 2 ZAKHAR KABLUCHKO 2.2. Gale coupling b etw een tw o Gaussian pro jections 6 2.3. Dualit y coupling for i.i.d. random unit v ectors 8 3. Tw o examples 9 3.1. Indep endence of face even ts 9 3.2. Exp ected face counts 10 4. Momen ts of random angles 11 4.1. Dualit y for moments of random angles 11 4.2. Momen ts of small order 12 4.3. Conjectures on high-dimensional random angles 13 5. A spherical Sylv ester problem 14 6. Limit theorems for face coun ts 16 6.1. Statemen t of the limit theorem 16 6.2. Bac kground from the theory of U -statistics 18 6.3. Pro of of Theorem 6.1: Represen tation of face coun t as U -statistic 19 6.4. Pro of of Theorem 6.1: Conditional exp ectation k ernels 20 6.5. Pro of of Theorem 6.1: The v ariance 23 6.6. Pro of of Theorem 6.1: Con volution op erator on the sphere and its eigenv alues 24 6.7. Pro of of Theorem 6.1: Completing the argumen t 27 6.8. Pro of of Theorem 6.1: Case k = 1 28 6.9. Exp onen tial concentration of face coun ts 29 App endix A. Basic facts on Gale duality 30 Ac knowledgemen t 33 References 33 1. Introduction 1.1. Random p olyhedral cones. Let U 1 , . . . , U n b e independent random v ectors, eac h distributed uniformly on the unit sphere S d − 1 in R d . The main ob ject of interest in the present pap er is the random p olyhedral cone given by the p ositive h ull of these vectors, W n,d := p os( U 1 , . . . , U n ) := { λ 1 U 1 + · · · + λ n U n : λ 1 ≥ 0 , . . . , λ n ≥ 0 } ⊆ R d . The p olar (conv ex dual) cone of W n,d is the intersection of the half-spaces with outer unit normal v ectors U 1 , . . . , U n , namely W ◦ n,d := { y ∈ R d : ⟨ x, y ⟩ ≤ 0 for all x ∈ W n,d } = n \ i =1 { y ∈ R d : ⟨ y , U i ⟩ ≤ 0 } . Th us, W ◦ n,d is the feasible set of a system of n random linear inequalities, making it a natural and fundamen tal ob ject. The random cones W n,d , W ◦ n,d , and related mo dels hav e b een studied b y Co ver and Efron [8], Donoho and T anner [11], and Hug and Sc hneider [20]; see also [15, 21, 22, 25, 43] and Chapters 5–6 of the b o ok by Schneider [44]. One imp ortant result is W endel’s form ula [49] RANDOM POL YHEDRAL CONES 3 (see also [45, Theorem 8.2.1]), which states that p ( n, d ) := P [ W n,d = R d ] = 1 2 n − 1 n − 1 X ℓ = d  n − 1 ℓ  . (1.1) Bey ond this, explicit formulas are kno wn for exp ectations of several geometric functionals of W n,d and W ◦ n,d (as w ell as of their conditioned versions). As one example, for ℓ ∈ { 0 , . . . , d − 1 } , let f ℓ ( W n,d ) denote the num b er of ℓ -dimensional faces of W n,d . Then Donoho and T anner [11, Theorem 1.6] sho w that E f ℓ ( W n,d ) =  n ℓ  p ( n − ℓ, n − d ) = 1 2 n − ℓ − 1  n ℓ  n − ℓ − 1 X p = n − d  n − ℓ − 1 p  . Other geometric functionals for whic h explicit exp ectation formulas are a v ailable include the conic in trinsic volumes (with the solid angle α ( W n,d ) as a sp ecial case), as well as sums of conic in trinsic v olumes ov er all faces of a given dimension and sums o ver the tangent cones at these faces. There also exist sp oradic results on second momen ts of certain geometric functionals [20, Theorem 8.1], but almost nothing seems to b e known about the full distribution of f ℓ ( W n,d ), α ( W n,d ), and related quan tities. 1.2. Main results. In the present pap er w e establish several distributional results for W n,d that go b eyond exp ectations and second momen ts. Moments of the solid angle. In Section 4 we study the solid angle of the cone W d,d (i.e., n = d ). F or a full-dimensional p olyhedral cone C ⊆ R d , its solid angle is deﬁned b y α ( C ) := P [ U ∈ C ], where U is uniformly distributed on the unit sphere in R d . W riting m ( d, k ) := E [ α ( W d,d ) k ] for the k -th moment, we pro ve the symmetry relation m ( d, k ) = m ( k , d ). As an application, we compute V ar( α ( W d,d )) = 2 − d ( d + 1) − 1 − 4 − d . W e also deriv e a closed-form expression for the third moment and prop ose a conjecture for the asymptotic b ehavior of m ( d, k ) as d → ∞ (with ﬁxed k ). Spheric al Sylvester pr oblem. In Section 5 we address a spherical analogue of Sylvester-t yp e ques- tions. F or n = d + 1 , d + 2 , d + 3 we compute the probability that W n,d ∩ S d − 1 is a spherical simplex. Moreo ver, for n = d + 2 we determine the distribution of the num b er of vertices of W d +2 ,d ∩ S d − 1 . High-dimensional limit the or ems for fac e c ounts. In Section 6 we prov e a distributional limit the- orem for face coun ts in a high-dimensional regime. Recall that f ℓ ( W n,d ) denotes the n umber of ℓ -dimensional faces of W n,d . W e consider n = d + k and ℓ = d − q , where k ∈ N and q ∈ N are ﬁxed, and let d → ∞ . W e sho w that the centered and normalized quantit y f d − q ( W d + k,d ) con v erges in distribution to a non-Gaussian limit; more precisely , d · f d − q ( W d + k,d )  d + k d − q  − p ( k + q , k ) ! w − → d →∞  k + q 2  k + q − 2 k − 1  2 k + q − 1 (1 − Q k ) , 4 ZAKHAR KABLUCHKO where Q k > 0 is an inﬁnitely divisible random v ariable with E Q k = 1 that admits the representation Q k d = X r =1 , 3 , 5 ,... Γ  k 2  Γ  r 2  π Γ  r + k 2  ! 2 Gamma  d r,k 2 , 1 2  , k ≥ 2 , and Q 1 d = Gamma( 1 2 , 1 2 ). Here Gamma( α, λ ) denotes a Gamma random v ariable with shap e pa- rameter α > 0 and rate parameter λ > 0, and all suc h v ariables are indep enden t. Moreo ver, d r,k :=  r + k − 1 r  −  r + k − 3 r − 2  is the dimension of the space of degree- r spherical harmonics on S k − 1 ⊆ R k for k ≥ 2. A more detailed statement is given in Theorem 6.1. Indep endenc e of fac e events. F or a subset I ⊆ { 1 , . . . , n } , deﬁne the cone F I := p os( U i : i ∈ I ), whic h is a candidate face of W n,d , and the even t A I := { F I is a face of W n,d } . Somewhat surprisingly , man y of the “face even ts” A I are independent. In Section 3.1 w e pro vide a suﬃcien t condition guaranteeing such independence. 1.3. Common technique: Gale duality. Although the results ab o ve may app ear unrelated at ﬁrst sigh t, their pro ofs share a common ingredien t: passing to the (linear) Gale transform. The necessary background on Gale transforms is collected in Section 2.1. The presen t work was inspired by F ric k et al. [14]. Their k ey observ ation is that the aﬃne Gale transform of n i.i.d. standard Gaussian p oints in R d ma y b e realized as n indep endent standard Gaussian p oints in R n − d − 1 , translated so that their barycenter is at the origin. Using this, they established an equiv alence b etw een the follo wing tw o problems: (a) determining the distribution of the Radon t yp e of d + 2 indep endent Gaussian p oints in R d , and (b) determining the lo cation of the sample mean among the order statistics of a one-dimensional i.i.d. Gaussian sample of size d + 2. F or further results on these t w o problems, see [4, 7, 23, 28, 50]. In contrast, the Gale duality input used in the present pap er is linear rather than aﬃne. The corresp onding results will b e stated in Prop osition 2.5 and Theorem 2.8. 1.4. Notation. Throughout the pap er w e use the following notation. F or a ﬁnite set A , let # A denote its cardinalit y . F or n ∈ N , write [ n ] : = { 1 , . . . , n } . Let R d b e d -dimensional Euclidean space with the standard inner pro duct ⟨· , ·⟩ and norm ∥ · ∥ . Let S d − 1 := { x ∈ R d : ∥ x ∥ = 1 } b e the unit sphere, and let e 1 , . . . , e d denote the standard orthonormal basis of R d . F or X ⊆ R d , we write pos( X ) for the p ositive hull of X and lin( X ) for the linear span of X , that is, p os( X ) := ( m X i =1 λ i x i : m ∈ N 0 , x 1 , . . . , x m ∈ X , λ 1 , . . . , λ m ≥ 0 ) , RANDOM POL YHEDRAL CONES 5 lin( X ) := ( m X i =1 λ i x i : m ∈ N 0 , x 1 , . . . , x m ∈ X , λ 1 , . . . , λ m ∈ R ) . If X = { x 1 , . . . , x n } is ﬁnite, w e also write p os( x 1 , . . . , x n ) and lin( x 1 , . . . , x n ) instead of p os( X ) and lin( X ). By con ven tion, p os( ∅ ) = lin( ∅ ) = { 0 } . A p olyhe dr al c one is the p ositive h ull of ﬁnitely many v ectors in R d . Equiv alently , it is an in tersection of ﬁnitely many half-spaces of the form { x ∈ R d : ⟨ x, u ⟩ ≤ 0 } , where u ∈ R d \ { 0 } . F or sequences ( a n ) n ∈ N and ( b n ) n ∈ N , we write a n ∼ b n if a n /b n → 1 as n → ∞ . F or random v ariables (or random elemen ts) X and Y , we write X d = Y to denote equality in distribution. W eak con vergence of random v ariables or elemen ts is denoted by w − → . 2. Linear Gale duality for random vector configura tions 2.1. Linear Gale dualit y. Gale duality (or the Gale transform) is a widely used to ol in conv ex and discrete geometry . F or detailed accounts of Gale dualit y and its applications, w e refer to the b o oks of Gr ¨ un baum [17, Section 5.4], Ziegler [52, Chapter 6], Matou ˇ sek [33, Section 5.6], McMullen and Shephard [36, Chapter 3], Bj¨ orner et al. [6, Section 3.4], Pineda Villavicencio [40, Section 2.14], McMullen [35, Sections 6B, 6C] and to the pap ers of McMullen [34], Shephard [48], and Gr¨ un baum and Shephard [18]. There are t wo versions of Gale duality: aﬃne and linear. The v ersion most commonly treated in the literature is the aﬃne one; in the presen t w ork, ho wev er, w e require linear Gale duality studied in [34, 48], which w e no w deﬁne. Deﬁnition 2.1 (Linear Gale duality) . Let d ′ , d ′′ ∈ N and put n := d ′ + d ′′ . Two vector conﬁgu- rations a 1 , . . . , a n ∈ R d ′ and b 1 , . . . , b n ∈ R d ′′ are said to b e in line ar Gale duality if the follo wing conditions hold: (i) the linear span of a 1 , . . . , a n is R d ′ and the linear span of b 1 , . . . , b n is R d ′′ ; (ii) if A ∈ R d ′ × n is the matrix with columns a 1 , . . . , a n and B ∈ R d ′′ × n is the matrix with columns b 1 , . . . , b n , then ev ery row of A is orthogonal to ev ery ro w of B . In matrix notation, AB ⊤ = 0 d ′ × d ′′ . Condition (i) states that rank A = d ′ and rank B = d ′′ . The row spaces of A and B , denoted b y Ro w ( A ) and Row( B ), are deﬁned as the linear subspaces in R n spanned b y the rows of A and B , resp ectiv ely . Condition (ii) states that Row( A ) and Ro w ( B ) are orthogonal to each other. In fact, b y the full rank assumption, they even form complementary orthogonal subspaces of R n . Thus conditions (i) and (ii) can b e stated as dim Ro w ( A ) = d ′ , dim Ro w ( B ) = d ′′ , Ro w ( A ) = Row( B ) ⊥ . (2.1) There is a pronounced corresp ondence b etw een the prop erties of linearly Gale-dual conﬁg- urations a 1 , . . . , a n and b 1 , . . . , b n . The follo wing Lemma 2.2 records a concrete instance of this corresp ondence: faces of the cone generated by the a i corresp ond to complements of p ositive span- ning subsets of the b j , and conv ersely . While this fact is standard and can b e found in some or other form in the references cited ab o ve, we include a complete elementary pro of in App endix A, Lemma A.4. 6 ZAKHAR KABLUCHKO Lemma 2.2 (F aces vs. p ositiv e spanning subsets under general linear position) . L et d ′ , d ′′ ∈ N and put n := d ′ + d ′′ . L et a 1 , . . . , a n ∈ R d ′ and b 1 , . . . , b n ∈ R d ′′ b e in line ar Gale duality, and assume that b oth c onﬁgur ations ar e in gener al line ar p osition. L et I ⊆ [ n ] satisfy # I ≤ d ′ − 1 . Then the fol lowing ar e e quivalent: (i) p os( a i : i ∈ I ) is a fac e of the p olyhe dr al c one p os( a 1 , . . . , a n ) . (ii) p os( b j : j ∈ I c ) = R d ′′ . Her e I c := [ n ] \ I is the c omplement of I . R emark 2.3 . Pairwise distinct v ectors v 1 , . . . , v m ∈ R d are said to b e in gener al line ar p osition if ev ery subset of { v 1 , . . . , v m } of size at most d is linearly indep endent. If m ≥ d , this is equiv alent to requiring that every d -elemen t subset of { v 1 , . . . , v m } is linearly indep endent. Example 2.4. Let I = ∅ . By con ven tion, p os( ∅ ) = { 0 } , so condition (i) says that { 0 } is a face of C A := p os( a 1 , . . . , a n ), i.e. that the cone C A is p ointed. If the v ectors a 1 , . . . , a n are in general linear position and linearly span R d ′ , then one c hecks that C A is either p oin ted or equal to R d ′ . Hence Lemma 2.2 with I = ∅ tak es the form p os( a 1 , . . . , a n )  = R d ′ ⇐ ⇒ p os( b 1 , . . . , b n ) = R d ′′ . 2.2. Gale coupling b etw een tw o Gaussian pro jections. Let G b e a Gaussian d 1 × d 2 matrix , i.e. a random matrix whose en tries are i.i.d. standard Gaussian random v ariables. W e view G as a random linear operator G : R d 2 → R d 1 , and refer to it as a Gaussian line ar op er ator . Its transp ose G ⊤ : R d 1 → R d 2 is again a Gaussian linear op erator. A k ey prop ert y is rotational in v ariance: for an y orthogonal transformations O 1 ∈ O ( d 1 ) and O 2 ∈ O ( d 2 ), the random matrix O 1 G O 2 has the same distribution as G . In particular, the notion of a Gaussian linear op erator do es not dep end on the choice of orthonormal bases in R d 1 and R d 2 . If d 2 ≥ d 1 , w e also refer to G as a Gaussian pr oje ction ; note, how ev er, that G is not a pro jection in the literal sense. Fix d ′ , d ′′ ∈ N and set n := d ′ + d ′′ . Let G : R n → R d ′ and H : R n → R d ′′ b e Gaussian pro jections. With probabilit y one, b oth Ker( G ) and Ker( H ) ⊥ are d ′′ -dimensional linear subspaces of R n (since G has rank d ′ and H has rank d ′′ almost surely). Let G ( n, d ′′ ) denote the Grassmannian of d ′′ -dimensional linear subspaces of R n , endow ed with its natural Borel σ -algebra. W e view Ker( G ) and Ker( H ) ⊥ as random elements of G ( n, d ′′ ). By rotational in v ariance of Gaussian matrices, these random subspaces are O ( n )-inv ariant: for every orthogonal transformation O ∈ O ( n ), O Ker( G ) d = Ker( G ) and O Ker( H ) ⊥ d = Ker( H ) ⊥ . Since there is a unique O ( n )-inv ariant probabilit y measure on G ( n, d ′′ ) (the Haar, or uniform, measure), it follows that both Ker( G ) and Ker( H ) ⊥ are distributed according to this me asure. So Ker( G ) d = Ker( H ) ⊥ , and this common distribution is uniform on G ( n, d ′′ ) . (2.2) The next prop osition shows that one can couple t w o Gaussian pro jections G and H on a common probabilit y space in suc h a wa y that the distributional iden tity (2.2) is realized by an almost sure equalit y of s ubspaces. ( G and H may b ecome dep enden t in this coupling.) Prop osition 2.5 (Coupling of Gaussian pro jections) . L et d ′ , d ′′ ∈ N and set n := d ′ + d ′′ . Ther e exists a pr ob ability sp ac e on which one c an deﬁne two Gaussian pr oje ctions G : R n → R d ′ and H : R n → R d ′′ such that Ker G = (Ker H ) ⊥ a.s. (2.3) RANDOM POL YHEDRAL CONES 7 Mor e over, if e 1 , . . . , e n denotes the standar d orthonormal b asis of R n , then for almost every r e al- ization of ( G , H ) the ve ctor c onﬁgur ations G e 1 , . . . , G e n ∈ R d ′ and H e 1 , . . . , H e n ∈ R d ′′ ar e in line ar Gale duality. W e shall give t wo pro ofs of Prop osition 2.5. The ﬁrst pro of is based on the follo wing general coupling principle. Lemma 2.6. L et X ∈ R p and Y ∈ R q b e r andom ve ctors, and let φ : R p → T and ψ : R q → T b e Bor el-me asur able maps into a Polish sp ac e T (e quipp e d with its Bor el σ -algebr a). Assume that φ ( X ) and ψ ( Y ) have the same distribution on T . Then ther e exists a pr ob ability sp ac e c arrying r andom ve ctors X ′ and Y ′ such that X ′ d = X , Y ′ d = Y , and φ ( X ′ ) = ψ ( Y ′ ) a.s. Pr o of. Let ( π X t ) t ∈ T b e a regular conditional distribution of X given φ ( X ), and let ( π Y t ) t ∈ T b e a regular conditional distribution of Y giv en ψ ( Y ). Existence of these k ernels follows since T is P olish; see, e.g., [12, T heorem 10.2.2]. By construction, for every t ∈ T , π X t  φ − 1 ( { t } )  = 1 and π Y t  ψ − 1 ( { t } )  = 1 . (2.4) No w construct a random elemen t Z with v alues in T such that Z d = φ ( X ) (equiv alently , Z d = ψ ( Y )). Conditionally on Z = t , sample X ′ according to π X t and Y ′ according to π Y t (for instance, take X ′ and Y ′ conditionally indep endent given Z ). Then X ′ d = X and Y ′ d = Y by the deﬁning prop ert y of regular conditional distributions. Moreov er, the support prop erties (2.4) imply φ ( X ′ ) = Z = ψ ( Y ′ ) almost surely . □ First pr o of of Pr op osition 2.5. W e apply Lemma 2.6 with T := G ( n, d ′′ ), X d = G and Y d = H , view ed as random elements of R d ′ × n and R d ′′ × n , resp ectively . Deﬁne Borel maps φ : R d ′ × n → T , φ ( M ) := Ker( M ) and ψ : R d ′′ × n → T , ψ ( M ) := Ker( M ) ⊥ . By (2.2), the random subspaces φ ( G ) = Ker( G ) and ψ ( H ) = Ker( H ) ⊥ ha ve the same distribution on T . Hence Lemma 2.6 yields a coupling of G and H on a common probability space such that Ker( G ) = Ker( H ) ⊥ a.s. (2.5) The matrices with columns G e 1 , . . . , G e n and H e 1 , . . . , H e n are precisely G and H . Since G and H are standard Gaussian matrices, they hav e full ro w rank almost surely , i.e. rank( G ) = d ′ and rank( H ) = d ′′ a.s. T ogether with (2.5) this sho ws that Conditions (2.1) are satisﬁed, hence the conﬁgurations G e 1 , . . . , G e n and H e 1 , . . . , H e n are in linear Gale dualit y . □ R emark 2.7 . The same argumen t prov es the follo wing more general claim. Consider random ma- trices A : R d ′ + d ′′ → R d ′ and B : R d ′ + d ′′ → R d ′′ suc h that, b oth, A and B ha ve full rank a.s. and Ker A has the same distribution as (Ker B ) ⊥ on the Grassmannian G ( d ′ + d ′′ , d ′′ ). Then, there is a coupling of A and B such that Ker A = (Ker B ) ⊥ a.s. The pro of follo ws from Lemma 2.6. 8 ZAKHAR KABLUCHKO Se c ond pr o of of Pr op osition 2.5. In this pro of, w e give an explicit construction of the coupling ( G , H ). Let X 1 , . . . , X d ′′ b e i.i.d. standard Gaussian p oints in R n = R d ′ + d ′′ . The linear Blaschk e– P etk antsc hin formula [45, Theorem 7.2.1] states that for ev ery nonnegative Borel function h : ( R n ) d ′′ → [0 , ∞ ), E h ( X 1 , . . . , X d ′′ ) = Z ( R n ) d ′′ e − 1 2 ( ∥ z 1 ∥ 2 + ... + ∥ z d ′′ ∥ 2 ) (2 π ) nd ′′ / 2 · h ( z 1 , . . . , z d ′′ ) d ′′ Y i =1 d z i = B Z G ( n,d ′′ ) Z L e − 1 2 ( ∥ z 1 ∥ 2 + ... + ∥ z d ′′ ∥ 2 ) (2 π ) nd ′′ / 2 · ( ∇ d ′′ ( z 1 , . . . , z d ′′ )) d ′ · h ( z 1 , . . . , z d ′′ ) d ′′ Y i =1 λ L (d z i ) ν n,d ′′ (d L ) , where ν n,d ′′ is the uniform probabilit y distribution on G ( n, d ′′ ), λ L is the Leb esgue measure on L ∈ G ( n, d ′′ ), ∇ d ′′ ( z 1 , . . . , z d ′′ ) denotes the d ′′ -dimensional volume of the parallelepip ed spanned b y the v ectors z 1 , . . . , z d ′′ , and B is a normalizing constan t whose v alue can b e found in [45, Eqn. (7.8), p. 271]. T o construct the coupling ( G , H ), w e consider the following tw o-stage random exp eriment. In a ﬁrst stage, construct a random Gaussian matrix G : R n → R d ′ . In a second stage, conditionally on Ker G = L , where L ⊆ R n is a d ′′ -dimensional linear subspace, w e construct random p oints ( Z 1 , . . . , Z d ′′ ) on L whose join t probability densit y with resp ect to ( λ L ) ⊗ d ′′ is given b y f L ( z 1 , . . . , z d ′′ ) := B · e − 1 2 ( ∥ z 1 ∥ 2 + ... + ∥ z d ′′ ∥ 2 ) (2 π ) nd ′′ / 2 · ( ∇ d ′′ ( z 1 , . . . , z d ′′ )) d ′ , z 1 , . . . , z d ′′ ∈ L. (The fact that f L in tegrates to 1 for every ﬁxed L ∈ G ( n, d ′′ ) follo ws from the Blaschk e–Petk antsc hin form ula with h ≡ 1 together with the fact that the integrand in R G ( n,d ′′ ) . . . ν n,d ′′ (d L ) do es not dep end on L .) With probability 1, the linear span of Z 1 , . . . , Z d ′′ is L . Now, for ev ery nonnegativ e Borel function h : ( R n ) d ′′ → [0 , ∞ ), the deﬁnition of Z 1 , . . . , Z d ′′ yields E h ( Z 1 , . . . , Z d ′′ ) = Z G ( n,d ′′ ) Z L B · e − 1 2 ( ∥ z 1 ∥ 2 + ... + ∥ z d ′′ ∥ 2 ) (2 π ) nd ′′ / 2 · ( ∇ d ′′ ( z 1 , . . . , z d ′′ )) d ′ · h ( z 1 , . . . , z d ′′ ) d ′′ Y i =1 λ L (d z i ) ν n,d ′′ (d L ) . So E h ( Z 1 , . . . , Z d ′′ ) = E h ( X 1 , . . . , X d ′′ ) for ev ery h and it follows that ( Z 1 , . . . , Z d ′′ ) hav e the same full joint distribution as ( X 1 , . . . , X d ′′ ), i.e. they are i.i.d. standard Gaussian p oints in R n . W e can no w construct H : R n → R d ′′ b y declaring that H ⊤ maps the standard orthonormal basis of R d ′′ to Z 1 , . . . , Z d ′′ . Then, H is a Gaussian matrix and (Ker H ) ⊥ = Im( H ⊤ ) = L = Ker G . □ 2.3. Dualit y coupling for i.i.d. random unit v ectors. The next result will b e rep eatedly used in the subsequen t sections. Theorem 2.8 (Main duality coupling) . Fix some d, k ∈ N . On a suitable pr ob ability sp ac e one c an c onstruct r andom ve ctors U 1 , . . . , U d + k , V 1 , . . . , V d + k such that (a) U 1 , . . . , U d + k ar e indep endent and uniform on the unit spher e S d − 1 in R d ; (b) V 1 , . . . , V d + k ar e indep endent and uniform on the unit spher e S k − 1 in R k ; RANDOM POL YHEDRAL CONES 9 (c) F or every outc ome in the underlying pr ob ability sp ac e and every set I ⊆ [ d + k ] with # I ≤ d − 1 we have p os( U i : i ∈ I ) is a fac e of p os( U 1 , . . . , U d + k ) ⇐ ⇒ pos( V i : i ∈ I c ) = R k . (2.6) Her e I c := [ d + k ] \ I is the c omplement of the index set I . Pr o of. Let e 1 , . . . , e d + k b e the standard orthonormal basis of R d + k . By Prop osition 2.5, we may couple Gaussian pro jections G : R d + k → R d and H : R d + k → R k suc h that, for almost every realization of ( G , H ), the v ectors G e 1 , . . . , G e d + k ∈ R d are in linear Gale dualit y with the vectors H e 1 , . . . , H e d + k ∈ R k . Moreo ver, with probabilit y 1, both conﬁgurations are in general linear p osition. After restricting to a full-measure even t w e ma y assume that these prop erties hold for al l outcomes in the probability space. W e can therefore apply Lemma 2.2 to the Gale-dual conﬁgurations ( G e i ) i ∈ [ d + k ] and ( H e i ) i ∈ [ d + k ] . It yields that for ev ery set I ⊆ [ d + k ] with # I ≤ d − 1, p os( G e i : i ∈ I ) is a face of p os( G e i : i ∈ [ d + k ]) ⇐ ⇒ p os( H e i : i ∈ I c ) = R k . (2.7) No w we deﬁne U i := G e i / ∥ G e i ∥ and V i := H e i / ∥ H e i ∥ for all i = 1 , . . . , d + k . (Note that G e i  = 0 and H e i  = 0 by the linear general position property .) Then (a) and (b) are satisﬁed. Also, for ev ery M ⊆ [ d + k ] we ha ve p os( G e i : i ∈ M ) = p os( U i : i ∈ M ) , p os( H e i : i ∈ M ) = p os( V i : i ∈ M ) . Substituting these iden tities into (2.7) yields (2.6), completing the pro of. □ R emark 2.9 . The coupling in Theorem 2.8 is naturally interpreted in the language of oriented matroids [6]. Indeed, for eac h outcome of the underlying probabilit y space, the vector conﬁgurations ( U 1 , . . . , U d + k ) ⊆ R d and ( V 1 , . . . , V d + k ) ⊆ R k determine oriented matroids on the common ground set [ d + k ], and the construction in the pro of yields that these oriented matroids are dual to one another (in the sense of Gale duality). In the present pap er, ho wev er, we do not make systematic use of oriented matroid theory: the only consequence of this duality that will b e needed is the corresp ondence stated in part (c) of Theorem 2.8. 3. Two examples Theorem 2.8 will serve as a key to ol throughout the remainder of the pap er. W e start b y illustrating its use with t wo simple applications. 3.1. Indep endence of face ev en ts. Consider the random cone W n,d = p os( U 1 , . . . , U n ), where U 1 , . . . , U n are independent and uniformly distrib uted on the unit sphere in R d . F or a subset I ⊆ [ n ] deﬁne the cone F I = p os( U i : i ∈ I ) and the random even t A I := { F I is a face of W n,d } . With probability 1, the vectors U 1 , . . . , U n are in general linear p osition (assuming d ≥ 2). Th us, for ℓ ∈ { 0 , . . . , d − 1 } , every ℓ -dimensional face of W n,d has the form F I for some I ⊆ [ n ] with 10 ZAKHAR KABLUCHKO # I = ℓ , but not every F I is necessarily a face of W n,d . In particular, the n umber of ℓ -dimensional faces of W n,d can b e represented as f ℓ ( W n,d ) = X I ⊆ [ n ] , # I = ℓ 1 A I . Somewhat unexp ectedly , man y of the face ev ents A I turn out to be indep endent; the next theorem mak es this precise. Theorem 3.1 (Indep endence of face even ts) . T ake d ≥ 3 and n ≥ d + 1 . L et I 1 , . . . , I r ⊆ [ n ] b e such that I s ∪ I t = [ n ] for al l 1 ≤ s  = t ≤ r and # I s ≤ d − 1 for al l s ∈ [ r ] . Then, the fac e events A I 1 , . . . , A I r ar e indep endent. Pr o of. W e use the coupling given in Theorem 2.8 with k = n − d . Recall that the random vectors V 1 , . . . , V n app earing in that theorem are indep endent and uniformly distributed on the unit sphere in R n − d . F or J ⊆ [ n ] consider the cone D J = p os( V j : j ∈ J ) ⊆ R n − d . By Theorem 2.8, A I = { F I is a face of W n,d } = { D I c = R n − d } for all I ⊆ [ n ] with # I ≤ d − 1 . (3.1) Under our assumptions, the index sets I c 1 , . . . , I c r are disjoin t. Hence the random ev ents { D I c 1 = R n − d } , . . . , { D I c r = R n − d } are indep endent. □ 3.2. Exp ected face coun ts. As a simple consequence of Theorem 2.8, we reco ver the following result of Donoho and T anner [11, Theorem 1.6]. Recall that W endel’s formula [49] (see also [45, Theorem 8.2.1]) states that p ( n, d ) = P [ W n,d = R d ] = 1 2 n − 1 n − 1 X ℓ = d  n − 1 ℓ  . (3.2) Prop osition 3.2 (Exp ected face num b ers) . F or every d ≥ 2 , n ≥ d + 1 and ℓ ∈ { 0 , . . . , d − 1 } , the exp e cte d numb er of ℓ -dimensional fac es of the r andom c one W n,d is E f ℓ ( W n,d ) =  n ℓ  p ( n − ℓ, n − d ) = 1 2 n − ℓ − 1  n ℓ  n − ℓ − 1 X r = n − d  n − ℓ − 1 r  . A lso, for every I ⊆ [ n ] with # I = ℓ ∈ { 0 , . . . , d − 1 } , we have P [ A I ] = p ( n − ℓ, n − d ) . Pr o of. F or ev ery I as ab ov e, F orm ula (3.1) gives P [ A I ] = P [p os( V i : i ∈ I c ) = R n − d ] = P [ W n − ℓ,n − d = R n − d ] = p ( n − ℓ, n − d ) , where the last step follows from W endel’s formula (3.2). T aking the sum ov er all subsets I ⊆ [ n ] with # I = ℓ gives the stated form ula for E f ℓ ( W n,d ). □ R emark 3.3 . T aking up a suggestion of Gale, Sc hneider [43] prop osed to study random p olytop es P obtained b y choosing their (aﬃne) Gale transform at random. (Throughout this remark, w e use the term “p olytop e” to refer to its c ombinatorial typ e ; only the combinatorial type is canonically determined b y the Gale transform.) The mo del of Schneider [43, p. 643] dep ends on an even probabilit y measure ϕ ; in the present remark we tak e ϕ to b e the uniform distribution on the unit sphere in R k . W e put n = d + k . Using the notation of Theorem 2.8, Schneider [43] considers a ( d − 1)-dimensional random p olytop e P with n v ertices whose (aﬃne) Gale transform is given by RANDOM POL YHEDRAL CONES 11 λ 1 V 1 , . . . , λ n V n ∈ R k (with suitable λ j > 0) conditioned on the even t pos( V 1 , . . . , V n ) = R k . By Example 2.4, this is the same conditioning even t as p os( U 1 , . . . , U n )  = R d . It then follo ws from Theorem 2.8 that the random cone W n,d = p os( U 1 , . . . , U n ) conditioned on the ev ent {W n,d  = R d } , can b e identiﬁed with the cone ov er P . Conv ersely , P can b e recov ered as a b ounded slice (cross- section) of this conditioned random cone. Such conditioned cones are called Cover–Efr on c ones in [20]. Consequen tly , the random p olytopes studied in Sc hneider [43] hav e the same distribution as the b ounded cross-sections of Co v er–Efron cones considered in [20]. (As noted in [43, Remark on p. 648], these t wo ob jects in particular hav e the same exp ected face num b ers.) 4. Moments of random angles 4.1. Dualit y for moments of random angles. The solid angle of a p olyhedral cone with non- empt y in terior C ⊆ R d is deﬁned as α ( C ) = ν d ( C ∩ S d − 1 ) ∈ [0 , 1] , where ν d is the uniform probabilit y distribution on S d − 1 . Let U 1 , . . . , U d b e i.i.d. random unit v ectors uniformly distributed on the unit sphere in R d , d ∈ N . The solid angle of the cone W d,d = p os( U 1 , . . . , U d ) is a random v ariable α d := α (p os( U 1 , . . . , U d )) taking v alues in the interv al [0 , 1 / 2]. W e are interested in the momen ts of α d , which w e denote by m ( d, k ) := E [ α k d ] , d, k ∈ N . Clearly , m ( d, 1) = E α d = 2 − d . (Indeed, the cones p os( ε 1 U 1 , . . . , ε d U d ), with ε j ∈ {± 1 } , hav e disjoin t interiors, co ver R d and hav e the same exp ected angle.) Theorem 4.1 (Duality for random solid angles) . F or al l d, k ∈ N we have m ( d, k ) = m ( k , d ) . Pr o of. Let U 1 , . . . , U d , U d +1 , . . . , U d + k b e i.i.d. random unit vectors with the uniform distribution on S d − 1 . W e shall compute the probability of the ev ent U d +1 , . . . , U d + k ∈ p os( U 1 , . . . , U d ) in tw o diﬀeren t wa ys. Primal metho d . Conditioning on U 1 , . . . , U d , we ha ve, by deﬁnition of the solid angle, P [ U d +1 , . . . , U d + k ∈ p os( U 1 , . . . , U d ) | U 1 , . . . , U d ] = ( α (p os( U 1 , . . . , U d ))) k . In tegrating ov er U 1 , . . . , U d w e arrive at P [ U d +1 , . . . , U d + k ∈ p os( U 1 , . . . , U d )] = E [( α (p os( U 1 , . . . , U d ))) k ] = m ( d, k ) . Dual metho d. By Theorem 2.8, we may realize ( U 1 , . . . , U d + k ) together with ( V 1 , . . . , V d + k ) on a common probabilit y space suc h that the dualit y (2.6) holds. In the follo wing we agree to ignore n ull ev ents. On the even t that p os( U 1 , . . . , U d ) ∩ S d − 1 is a spherical simplex (whic h holds a.s.), the con- dition U d +1 , . . . , U d + k ∈ p os( U 1 , . . . , U d ) is equiv alent to requiring that all facets of p os( U 1 , . . . , U d ) remain facets of p os( U 1 , . . . , U d + k ). Hence, P h U d +1 , . . . , U d + k ∈ p os( U 1 , . . . , U d ) i = P h ∀ ℓ ∈ [ d ] : p os( U i : i ∈ [ d ] \ { ℓ } ) is a facet of pos( U 1 , . . . , U d + k ) i = P h ∀ ℓ ∈ [ d ] : p os( V i : i ∈ ([ d ] \ { ℓ } ) c ) = R k i , 12 ZAKHAR KABLUCHKO where we used (2.6). Since ([ d ] \ { ℓ } ) c = { ℓ, d + 1 , . . . , d + k } , this b ecomes P [ U d +1 , . . . , U d + k ∈ p os( U 1 , . . . , U d )] = P h ∀ ℓ ∈ [ d ] : p os( V ℓ , V d +1 , . . . , V d + k ) = R k i . No w, w e observ e that p os( V ℓ , V d +1 , . . . , V d + k ) = R k if and only if V ℓ ∈ − In t p os( V d +1 , . . . , V d + k ), ignoring null ev ents. Th us, P [ U d +1 , . . . , U d + k ∈ p os( U 1 , . . . , U d )] = P [ V 1 , . . . , V d ∈ − p os( V d +1 , . . . , V d + k )] = E [( α (p os( V d +1 , . . . , V d + k ))) d ] = m ( k , d ) . Comparing the results obtained b y b oth metho ds, we conclude that m ( d, k ) = m ( k , d ). □ 4.2. Momen ts of small order. Theorem 4.1 allo ws us to ﬁnd explicit form ulas for the ﬁrst three momen ts of α d . Theorem 4.2 (Momen ts of random angles) . F or every d ∈ N , the ﬁrst thr e e moments of the r andom variable α d := α (p os( U 1 , . . . , U d )) ar e given by E [ α d ] = m ( d, 1) = m (1 , d ) = 2 − d , (4.1) E [ α 2 d ] = m ( d, 2) = m (2 , d ) = 1 2 d ( d + 1) , V ar[ α d ] = 1 2 d ( d + 1) − 1 4 d , (4.2) E [ α 3 d ] = m ( d, 3) = m (3 , d ) = 1 (4 π ) d E [ S d ] = 1 (4 π ) d Z 2 π 0 x d f S ( x )d x, (4.3) wher e S is a r andom variable with pr ob ability density f S ( x ) = − ( x 2 − 4 π x + 3 π 2 − 6) cos x − 6( x − 2 π ) sin x − 2( x 2 − 4 π x + 3 π 2 + 3) 16 π cos 4 ( x/ 2) , 0 ≤ x ≤ 2 π . Pr o of. T o giv e an alternative pro of that E [ α d ] = 2 − d , observ e that m (1 , d ) = 2 − d since V 1 is uniform on {± 1 } and hence p os( V 1 ) is a ray with angle 1 / 2. T o prov e (4.2), it suﬃces to compute m (2 , d ). If V 1 , V 2 are indep endent and uniformly dis- tributed on the unit circle in R 2 , then the random angle α (p os( V 1 , V 2 )) is uniformly distributed on [0 , 1 / 2]. The d -th momen t of this random v ariable is 2 − d · 1 / ( d + 1), and the claim follo ws. Finally , the formula for m (3 , d ) follo ws from the follo wing non-trivial fact due to Crofton and Exh umatus [9]: The spherical area S of the random spherical triangle with vertices V 1 , V 2 , V 3 that are indep endent and uniform on S 2 has probabilit y densit y f S giv en as ab o ve. Pro ofs can b e found in [39] and [13]; see also [46] and [37]. Then α (p os( V 1 , V 2 , V 3 )) = S/ (4 π ) and m (3 , d ) = E [ S d / (4 π ) d ]. □ R emark 4.3 (The third moment) . Explicit formulas for E [ S d ] with d ≤ 10 are given in [46]. (In the form ula for E [ S 4 ], a factor of 108 is missing in front of ζ (3).) These form ulas giv e the trivial v alues E [ α 3 1 ] = 1 8 , E [ α 3 2 ] = 1 32 (since α 2 is uniform on [0 , 1 / 2]) as well as E [ α 3 3 ] = 3 128 − 3 log 2 16 π 2 , E [ α 3 4 ] = π 4 − 6 π 2 log 2 − 27 ζ (3) 64 π 4 , E [ α 3 5 ] = 5 512 − 15 log 2 128 π 2 . RANDOM POL YHEDRAL CONES 13 R emark 4.4 (Asymptotics of the third momen ts) . Since f S (2 π ) = 12 − π 2 16 π > 0 and f S ( x ) is con tinuous at x = 2 π , the standard Laplace asymptotics gives E [ α 3 d ] = 1 (4 π ) d E [ S d ] ∼ 1 (4 π ) d f S (2 π ) (2 π ) d +1 d + 1 ∼ 12 − π 2 8 1 2 d d , d → ∞ . R emark 4.5 . In this section, we considered the cone W n,d with n = d only . Our metho ds do not allo w to compute the v ariance of α ( W n,d ) for n > d . What we can show is that for n ≥ d , E [ α 2 ( W n,d )] = E [ f 2 1 ( W n +2 ,n +2 − d )] − E [ f 1 ( W n +2 ,n +2 − d )] ( n + 2)( n + 1) . (4.4) Pr o of of (4.4) . Let k ∈ N . By Theorem 2.8, w e may realize i.i.d. random v ectors U 1 , . . . , U n + k ∼ Unif ( S d − 1 ) together the i.i.d. random vectors V 1 , . . . , V n + k ∼ Unif ( S n + k − d − 1 ) on a common proba- bilit y space such that the duality (2.6) holds. W e ha v e E [ α k ( W n,d )] = P [ U n +1 , . . . , U n + k ∈ − p os( U 1 , . . . , U n )] = P h p os( U 1 , . . . , U n , U n + i ) = R d ∀ i = 1 , . . . , k i . Applying the dualit y (2.6), with U i ’s and V j ’s interc hanged, to the right-hand side gives E [ α k ( W n,d )] = P  p os  { V n +1 , . . . , V n + k } \ { V n + i }  is a face of p os( V 1 , . . . , V n + k ) ∀ i = 1 , . . . , k  . If k = 2, the condition on the right-hand side states that p os( V n +1 ) and pos( V n +2 ) are faces of p os( V 1 , . . . , V n +2 ). So, using exchangeabilit y , E [ α 2 ( W n,d )] = P [pos( V n +1 ) , pos( V n +2 ) ∈ F 1 (p os( V 1 , . . . , V n +2 ))] = 1 ( n + 2)( n + 1) E X i  = j i,j ∈{ 1 ,...,n +2 } 1 { pos( V i ) ∈F 1 (pos( V 1 ,...,V n +2 )) } 1 { pos( V j ) ∈F 1 (pos( V 1 ,...,V n +2 )) } = 1 ( n + 2)( n + 1) E  f 2 1 (p os( V 1 , . . . , V n +2 )) − f 1 (p os( V 1 , . . . , V n +2 ))  , and the pro of is complete since p os( V 1 , . . . , V n +2 ) has the same la w as W n +2 ,n +2 − d . □ 4.3. Conjectures on high-dimensional random angles. It seems diﬃcult to compute m ( d, k ) = m ( k , d ) with k , d ≥ 4. Here we state a conjecture on the asymptotics of m ( d, k ) as d → ∞ with ﬁxed k ≥ 2. Let V 1 , . . . , V k b e indep enden t and uniform on S k − 1 . The maximal p ossible v alue of the random v ariable α (p os( V 1 , . . . , V k )) is 1 / 2. An analysis of conﬁgurations of k unit v ectors in R k close to b eing linearly dep enden t suggests that the density of α (p os( V 1 , . . . , V k )) at 1 / 2 should b e p ositive, which (arguing as in Remark 4.4) leads to the follo wing conjecture. Conjecture 4.6 (Asymptotics for higher moments) . F or every ﬁxe d k ∈ { 2 , 3 , . . . } , we have m ( d, k ) ∼ c k / (2 d · d ) as d → ∞ , wher e c k > 0 is a c onstant. F or example, c 2 = 1 and c 3 = (12 − π 2 ) / 8 . (Note that the c ase k = 1 is diﬀer ent sinc e m ( d, 1) = 2 − d .) This b eha vior of moments suggests that, for large d , the random v ariable α d = α (p os( U 1 , . . . , U d )) can b e approximated b y a mixture of a random “bac kground” of size 2 − d and rare random “spikes” with size of order 1 and probabilit y of order 1 / ( d · 2 d ). More precisely , let A d b e an ev ent with P [ A d ] = 1 / ( d · 2 d ) and let X , Y ≥ 0 b e random v ariables with ﬁnite momen ts. Then, the random 14 ZAKHAR KABLUCHKO v ariables Z d := (1 − 1 A d )2 − d X + 1 A d Y ha ve moments whose large- d b ehavior is similar to that of α d , namely E [ Z d ] ∼ 2 − d E [ X ] as d → ∞ and E [ Z k d ] ∼ E [ Y k ] / ( d · 2 d ) for k ≥ 2. This suggests the follo wing conjectures. Conjecture 4.7 (W eak limit for random high-dimensional angle) . As d → ∞ , the r andom variables 2 d α d c onver ge in distribution to some r andom variable X ≥ 0 with E [ X ] = 1 . Conjecture 4.8 (Rare spik es for high-dimensional random angle) . The se quenc e of me asur es ( µ d ) d ∈ N with µ d ( A ) := 2 d d · P [ α d ∈ A ] , A ⊆ (0 , ∞ ) , c onver ges we akly to a non-trivial limit. 5. A spherical Syl vester problem T ake some d ≥ 2, k ∈ N , and let U 1 , . . . , U d + k b e i.i.d. random unit vectors uniformly dis- tributed on the unit sphere in R d . W e consider the random spherical p olytop e P d + k,d := W d + k,d ∩ S d − 1 = p os( U 1 , . . . , U d + k ) ∩ S d − 1 . W endel’s form ula (1.1) gives the probability that P d + k,d is the whole sphere. Prop osition 5.1 (Probabilit y of spherical simplex) . F or d ≥ 2 , k ∈ N , the pr ob ability that P d + k,d is a spheric al simplex (with d vertic es) e quals  d + k d  m ( d, k ) =  d + k k  m ( k , d ) . Pr o of. The ev ent {P d + k,d is a spherical simplex } o ccurs if and only if some (necessarily unique, for d ≥ 2) set of k v ectors U i 1 , . . . , U i k is contained in the p ositive h ull of the remaining d v ectors. By exc hangeability , it follows that P [ P d + k,d is a spherical simplex] =  d + k d  · P [ U d +1 , . . . , U d + k ∈ p os( U 1 , . . . , U d )] =  d + k d  · E [( α (p os( U 1 , . . . , U d ))) k ] =  d + k d  m ( d, k ) , and the pro of is complete since m ( d, k ) = m ( k , d ) b y Theorem 4.1. □ Example 5.2. T aking k = 1 , 2 , 3 and recalling Theorem 4.2 yields the following results: P [ P d +1 ,d is a spherical simplex] = d + 1 2 d , (5.1) P [ P d +2 ,d is a spherical simplex] = d + 2 2 d +1 , (5.2) P [ P d +3 ,d is a spherical simplex] = 1 (4 π ) d  d + 3 3  Z 2 π 0 x d f S ( x )d x, (5.3) where f S ( x ) is as in Theorem 4.2. Equation (5.1) reco vers a form ula obtained in [31, Lemma 4.4]; see also [30, Theorem 3.6], [32, Corollary 8.1], [23, Section 4.2]. Note that (5.1) admits a simple pro of: P d +1 ,d is a spherical simplex if one of the v ectors ( d + 1 c hoices) is inside the p ositive hull of the remaining vectors, and the probability that U d +1 ∈ p os( U 1 , . . . , U d ) is the exp ected angle of p os( U 1 , . . . , U d ), which is equal to 2 − d for symmetry reasons (see [31, Section 4] for details). RANDOM POL YHEDRAL CONES 15 On the other hand, Equations (5.2) and (5.3) seem to b e new. F or example, (5.3) com bined with Remark 4.3 giv es P [ P 6 , 3 is a spherical simplex] = 15 32 − 15 log 2 4 π 2 ≈ 0 . 205385 , P [ P 7 , 4 is a spherical simplex] = 35 64 − 105 log 2 32 π 2 − 945 ζ (3) 64 π 4 ≈ 0 . 134219 , P [ P 8 , 5 is a spherical simplex] = 35 64 − 105 log 2 16 π 2 ≈ 0 . 085987 . Next we consider the case n = d + 2 in more detail. The n umber of vertices of the spherical p olytop e P d +2 ,d can take v alues d (if P d +2 ,d is a spherical simplex), d + 1, d + 2 and 0 (equiv alen tly , P d +2 ,d = S d − 1 ). In the next theorem we compute the probabilities of these p ossibilities. Theorem 5.3 ( d + 2 p oints on S d − 1 ) . F or al l d ≥ 2 we have P [ P d +2 ,d = S d − 1 ] = d + 2 2 d +1 , (5.4) P [ P d +2 ,d is a spheric al simplex ] = d + 2 2 d +1 , (5.5) P [ P d +2 ,d has d + 1 vertic es ] = ( d − 2)( d + 2) 2 d +1 , (5.6) P [ P d +2 ,d has d + 2 vertic es ] = 1 − d ( d + 2) 2 d +1 . (5.7) Pr o of. By W endel’s form ula (3.2), P [ P d +2 ,d = S d − 1 ] = p ( d + 2 , d ) = ( d + 2) / 2 d +1 , whic h pro ves (5.4). W e already prov ed (5.5) in Example 5.2. It remains to prov e (5.6) and (5.7). By Prop osition 3.2 with ℓ = 1 (taking into accoun t that f 0 ( P d +2 ,d ) = f 1 ( W d +2 ,d )), E [ d + 2 − f 0 ( P d +2 ,d )] = ( d + 2)(1 − p ( d + 1 , 2)) = ( d + 2) p ( d + 1 , d − 1) = ( d + 2)(2 d + 2) 2 d +1 , where w e again used W endel’s form ula (3.2). On the other hand, by deﬁnition of exp ectation and (5.5), E [ d + 2 − f 0 ( P d +2 ,d )] = ( d + 2) · P [ P d +2 ,d = S d − 1 ] + 2 · P [ P d +2 ,d is a simplex] + 1 · P [ P d +2 ,d has d + 1 vertices] = ( d + 2)( d + 2 + 2) 2 d +1 + P [ P d +2 ,d has d + 1 vertices] . Comparing both results giv es (5.6). T o pro ve (5.7), observ e that the complement of {P d +2 ,d has d + 2 vertices } is the disjoin t union of the ev ents app earing in (5.4), (5.5), (5.6). □ Example 5.4 (5 p oin ts on S 2 ) . T aking d = 3 in Theorem 5.3 giv es: The probability that the spherical conv ex h ull of ﬁve random p oints on S 2 , c hosen uniformly and indep endently , is a spherical triangle, quadrilateral, or pentagon is 5 / 16 , 5 / 16 , 1 / 16, resp ectiv ely . (Also, with probabilit y 5 / 16, their p ositive h ull is the whole of R 3 ). Elemen tary pro ofs can b e found in [30, Theorem 3.6], [32, Exercises 8.1, 8.2 on p. 170], [46]. Let us now consider the regime where d → ∞ while k ≥ 2 is ﬁxed. Conjecture 4.6, together with Prop osition 5.1, w ould imply that the probabilit y that P d + k,d is a spherical simplex is asymp- totically equiv alent to ( c k /k !) d k − 1 2 − d for some constan t c k > 0. Section 4.3 suggests the following 16 ZAKHAR KABLUCHKO mec hanism by whic h this rare ev ent ma y occur: a subset of d vectors U i 1 , . . . , U i d spans a cone with solid angle of constant order, and the remaining k v ectors fall inside this cone. This leads to the follo wing conjecture. Conjecture 5.5. Fix k ≥ 2 . The c onditional distribution of α ( W d + k,d ) , given that W d + k,d ∩ S d − 1 is a spheric al simplex, c onver ges we akly (as d → ∞ ) to a limit law not c onc entr ate d at 0 . Let us record one further consequence of Theorem 2.8. Prop osition 5.6 (Probability of ℓ -neighborliness) . L et d, k ∈ N . L et U 1 , . . . , U d + k b e indep endent and uniform on S d − 1 and let V 1 , . . . , V d + k b e indep endent and uniform on S k − 1 . F or every ℓ ∈ { 0 , . . . , d − 1 } , the fol lowing events have the same pr ob ability: (a) f ℓ ( W d + k,d ) =  d + k ℓ  , that is, for every subset I ⊆ [ d + k ] with # I = ℓ , the c one p os( U i : i ∈ I ) is an ℓ -dimensional fac e of W d + k,d = p os( U 1 , . . . , U d + k ) . (b) F or every J ⊆ [ d + k ] with # J = d + k − ℓ , one has T j ∈ J HS j = ∅ , wher e HS j := { x ∈ S k − 1 : ⟨ x, V j ⟩ ≤ 0 } , j = 1 , . . . , d + k . Equivalently, the family of half-spher es (HS j ) d + k j =1 c overs every p oint of S k − 1 at most d + k − ℓ − 1 times. (c) The family (HS j ) d + k j =1 c overs every p oint of S k − 1 at le ast ℓ + 1 times. Pr o of. By Theorem 2.8 (c), the ev ent in (a) has the same probability as the even t that for every J ⊆ [ d + k ] with # J = d + k − ℓ one has p os( V j : j ∈ J ) = R k . By polarity , p os( V j : j ∈ J )  = R k holds if and only if there exists x ∈ S k − 1 with ⟨ x, V j ⟩ ≤ 0 for all j ∈ J , that is, if and only if T j ∈ J HS j  = ∅ . Hence p os( V j : j ∈ J ) = R k is equiv alent to T j ∈ J HS j = ∅ , which is precisely (b). T o see that the even ts in (b) and (c) ha ve equal probabilities, observe that Ev ent in (b) ⇐ ⇒ ∀ x ∈ S k − 1 : # { j ∈ [ d + k ] : x ∈ HS j } ≤ d + k − ℓ − 1 ⇐ ⇒ ∀ x ∈ S k − 1 : # { j ∈ [ d + k ] : x ∈ S k − 1 \ HS j } ≥ ℓ + 1 . Since S k − 1 \ HS j = { z ∈ S k − 1 : ⟨ z , − V j ⟩ < 0 } and since ( V 1 , . . . , V d + k ) and ( − V 1 , . . . , − V d + k ) ha ve the same distribution, the probabilit y of the latter ev ent equals the probability of (c). □ 6. Limit theorems for f a ce counts 6.1. Statemen t of the limit theorem. Recall that w e consider the random p olyhedral cone W n,d = p os( U 1 , . . . , U n ), where U 1 , . . . , U n are indep enden t and uniformly distributed on the unit sphere in R d . Not m uch seems to b e known ab out limit distributions of f ℓ ( W n,d ) as n, d, ℓ → ∞ . W e no w describe the asymptotic ﬂuctuations of the ( d − q )-dimensional face count of W d + k,d in the regime d → ∞ with k ∈ N and q ∈ N ﬁxed. Recall from Prop osition 3.2 the explicit formula E f d − q ( W d + k,d ) =  d + k d − q  p ( k + q , k ) = 1 2 k + q − 1  d + k d − q  k + q − 1 X r = k  k + q − 1 r  , d ≥ q . Theorem 6.1 (Distributional limit for face counts) . L et k ∈ N and q ∈ N b e ﬁxe d. Then, as d → ∞ , the numb er of ( d − q ) -dimensional fac es of W d + k,d satisﬁes d · f d − q ( W d + k,d )  d + k d − q  − p ( k + q , k ) ! w − → d →∞  k + q 2  k + q − 2 k − 1  2 k + q − 1 (1 − Q k ) , (6.1) RANDOM POL YHEDRAL CONES 17 wher e Q k > 0 is a r andom variable admitting the distributional r epr esentation Q 1 d = Gamma( 1 2 , 1 2 ) (for k = 1 ) and Q k d = X n =1 , 3 , 5 ,... Γ  k 2  Γ  n 2  π Γ  n + k 2  ! 2 Gamma  d n,k 2 , 1 2  , for k ≥ 2 . (6.2) Her e d n,k :=  n + k − 1 n  −  n + k − 3 n − 2  with k ≥ 2 , n ∈ N is the dimension of the sp ac e of de gr e e n spheric al harmonics on S k − 1 (se e Se ction 6.6), and Gamma( α, λ ) denotes a Gamma-variable with shap e p ar ameter α > 0 , r ate p ar ameter λ > 0 , and al l such variables ar e indep endent. The exp e ctation and varianc e of Q k ar e given by E Q k = 1 , V ar Q k = 4 π 2 ψ 1  k 2  , (6.3) wher e ψ 1 is the trigamma function, deﬁne d for x > 0 by ψ 1 ( x ) := d 2 d x 2 log Γ( x ) = P ∞ j =0 1 ( j + x ) 2 . Final ly, as d → ∞ we have V ar f d − q ( W d + k,d )  d + k d − q  ! ∼  k + q 2  2  k + q − 2 k − 1  2 2 2 k +2 q − 2 V ar Q k d 2 =  k + q 2  2  k + q − 2 k − 1  2 ψ 1  k 2  2 2 k +2 q − 4 π 2 · d 2 . (6.4) Example 6.2 ( k = 2) . F or k = 2, we ha ve d n, 2 = 2 for n ≥ 1. The distributional represen ta- tion (6.2) simpliﬁes to Q 2 d = 8 π 2 X n =1 , 3 , 5 ,... E n n 2 , where E 1 , E 3 , E 5 , . . . are i.i.d. exp onen tial with unit mean. It follo ws that the Laplace transform of Q 2 is given b y E  e − sQ 2  = Y n =1 , 3 , 5 ,... E  exp  − s 8 π 2 E n n 2  = Y n =1 , 3 , 5 ,... 1 1 + 8 s π 2 n 2 = 1 cosh  √ 2 s  , s ≥ 0 . This distribution is well known and its prop erties hav e b een review ed in [5]. In particular, the follo wing in terpretation is known. Let ( B t ) t ≥ 0 b e standard Bro wnian motion with B 0 = 0, and deﬁne the ﬁrst exit time from ( − 1 , 1) by τ 1 := inf { t ≥ 0 : | B t | = 1 } . Then τ 1 has the same la w as Q 2 . Let us record a con vex-dual analogue of Theorem 6.1. Let U 1 , . . . , U n b e indep endent random v ectors, each uniformly distributed on S d − 1 , where n > d . The hyperplanes U ⊥ 1 , . . . , U ⊥ n induce a conical tessellation of R d in to p olyhedral cones. The num b er of cones is almost surely constan t and equals C ( n, d ) = 2 P d − 1 ℓ =0  n − 1 ℓ  , b y the Steiner–Sc hl¨ aﬂi form ula; see [45, Lemma 8.2.1]. Cho ose one of these cones uniformly at random and denote it by S n,d . The random cone S n,d is called the r andom Schl¨ aﬂi c one ; see [20, Deﬁnition 3.2]. Corollary 6.3 (Distributional limit for face counts of Sc hl¨ aﬂi cones) . Fix k ∈ N and q ∈ N . Then, as d → ∞ , the numb er of q -dimensional fac es of S d + k,d satisﬁes d · f q ( S d + k,d )  d + k d − q  − p ( k + q , k ) ! w − → d →∞  k + q 2  k + q − 2 k − 1  2 k + q − 1 (1 − Q k ) , (6.5) 18 ZAKHAR KABLUCHKO wher e Q k > 0 is the same r andom variable as in The or em 6.1. Pr o of. Let W ◦ d + k,d denote the con vex dual of W d + k,d . Then f q ( W ◦ d + k,d ) = f d − q ( W d + k,d ), and Theorem 6.1 can b e rewritten as d · f q ( W ◦ d + k,d )  d + k d − q  − p ( k + q , k ) ! w − → d →∞  k + q 2  k + q − 2 k − 1  2 k + q − 1 (1 − Q k ) . (6.6) By [20, Theorem 3.1], S d + k,d has the same distribution as W ◦ d + k,d conditioned on the ev ent {W d + k,d  = R d } . Moreov er, W endel’s formula (1.1) yields P [ W d + k,d  = R d ] = 1 − p ( d + k , d ) = 1 − 1 2 d + k − 1 d + k − 1 X ℓ = d  d + k − 1 ℓ  → 1 as d → ∞ . Hence conditioning on {W d + k,d  = R d } b ecomes asymptotically negligible, and w e may replace f q ( W ◦ d + k,d ) by f q ( S d + k,d ) in (6.6) without aﬀecting conv ergence in distribution. □ 6.2. Bac kground from the theory of U - statistics. The pro of of Theorem 6.1 relies on the theory of U -statistics. In this section, we introduce the necessary notation and recall several classical results on U -statistics; see the b o oks of Lee [29], Korolyuk and Boro vskikh [27], and Serﬂing [47, Chapter 5] for further background. Let X 1 , X 2 , . . . b e i.i.d. random elemen ts in a measurable space ( X , A ) with common law µ . Let h : X m → R b e a kernel of order m ∈ N . Our standing assumption is that h is measurable, symmetric (inv arian t under p erm utations of its arguments), and E  h ( X 1 , . . . , X m ) 2  < ∞ . F or n ≥ m , the U -statistic asso ciated with h is U n :=  n m  − 1 X 1 ≤ i 1 < ··· 0 . Equiv alently , g r ( x 1 , . . . , x r ) = g 0 for all r ∈ { 1 , . . . , c − 1 } ( µ r -a.e.), but g c ( X 1 , . . . , X c ) is not a.s. constan t. If h is degenerate of order c , Equation (6.7) implies (see, e.g., [47, Section 5.3.4]) V ar U n =  m c  n − m m − c   n m  ζ c + o ( n − c ) ∼  m c  2 c ! n c ζ c , n → ∞ . (6.8) The next classical theorem c haracterizes the weak limit of the U -statisic U n under degeneracy of order 2; see [29, Section 3.2.2], [27, Chapter 4], [42, p. 168] or [47, Section 5.5.2, p. 194]. Theorem 6.4 (Non-Gaussian weak limit under degeneracy of order 2) . Assume h : X m → R is de gener ate of or der 2 (e quivalently, ζ 1 = 0 and ζ 2 > 0 ). Deﬁne the (self-adjoint, Hilb ert–Schmidt, henc e c omp act) op er ator T : L 2 ( X , µ ) → L 2 ( X , µ ) by ( T f )( x ) := Z X e g 2 ( x, y ) f ( y ) µ (d y ) . L et λ 1 , λ 2 , . . . b e the (r e al) non-zer o eigenvalues of T , liste d with multiplicities, and let ξ 1 , ξ 2 , . . . b e i.i.d. standar d normal r andom variables. Then n ( U n − EU n ) w − → n →∞  m 2  ∞ X j =1 λ j ( ξ 2 j − 1) . The series P ∞ j =1 λ j ( ξ 2 j − 1) con verges in L 2 and a.s. (since P ∞ j =1 λ 2 j < ∞ b y the Hilb ert– Sc hmidt prop erty of T and E [ ξ 4 j ] < ∞ ) and hence deﬁnes a prop er random limit. 6.3. Pro of of Theorem 6.1: Represen tation of face coun t as U -statistic. Sections 6.3 – 6.8 are devoted to the pro of of Theorem 6.1. The ﬁrst step is to in terpret f ℓ ( W n,d ) /  n ℓ  as a U -statistic, whic h becomes possible after passing to the Gale dual. Consider the random cone W n,d with n > d ≥ 2 and let ℓ ∈ { 0 , . . . , d − 1 } . With probabilit y 1, the v ectors U 1 , . . . , U n spanning this cone are in general linear p osition. Thus, ev ery ℓ -dimensional face of W n,d has the form pos( U i : i ∈ I ) for some unique set I ⊆ [ n ] with # I = ℓ and hence f ℓ ( W n,d ) = X I ⊆ [ n ] # I = ℓ 1 { pos( U i : i ∈ I ) is a face of W n,d } . W e now use the coupling given in Theorem 2.8 (with k = n − d ). Recall that the random vectors V 1 , . . . , V n app earing in that theorem are indep endent and uniformly distributed on the unit sphere S n − d − 1 ⊆ R n − d . By Theorem 2.8, we hav e the follo wing equality of random ev ents: { p os( U i : i ∈ I ) is a face of W n,d } = { p os( V j : j ∈ I c ) = R n − d } , for all I ⊆ [ n ] with # I ≤ d − 1. Putting J := I c and dividing by  n ℓ  w e arriv e at the represen tation f ℓ ( W n,d )  n ℓ  =  n n − ℓ  − 1 X J ⊆ [ n ] # J = n − ℓ 1 { pos( V j : j ∈ J )= R n − d } , ℓ ∈ { 0 , . . . , d − 1 } . (6.9) 20 ZAKHAR KABLUCHKO The righ t-hand side is a U -statistic with the sample space X = S n − d − 1 (endo wed with the uniform distribution ν n − d ) and the kernel h : X n − ℓ → { 0 , 1 } giv en by h ( v 1 , . . . , v n − ℓ ) = 1 { pos( v 1 ,...,v n − ℓ )= R n − d } , v 1 , . . . , v n − ℓ ∈ S n − d − 1 . Recall that in Theorem 6.1 w e consider f ℓ ( W n,d ), where the num b er of p oin ts is n = d + k , the dimension of faces is ℓ = d − q , and the parameters k ∈ N and q ∈ N are ﬁxed. Let us summarize our setting: • The sample space is X = S k − 1 , endow ed with the uniform distribution ν k . • V 1 , . . . , V n , where n = d + k , are i.i.d. random vectors uniformly distributed on S k − 1 . • The order of the U -statistic is m := n − ℓ = k + q and the kernel h : X m → { 0 , 1 } is h ( v 1 , . . . , v m ) = 1 { pos( v 1 ,...,v m )= R k } , v 1 , . . . , v m ∈ S k − 1 . By (6.9) we ha ve the identiﬁcation U n =  d + k m  − 1 X 1 ≤ i 1 < ··· 0. Recall from Section 6.2 that for c ∈ { 1 , . . . , m } and unit vectors v 1 , . . . , v c ∈ S k − 1 , w e consider the conditional exp ectation kernels g c ( v 1 , . . . , v c ) = E h ( v 1 , . . . , v c , V c +1 , . . . , V m ) = P [p os( v 1 , . . . , v c , V c +1 , . . . , V m ) = R k ] . W endel’s form ula (1.1) gives g 0 = E h ( V 1 , . . . , V m ) = P [p os( V 1 , . . . , V m ) = R k ] = p ( m, k ) , EU n = g 0 = p ( m, k ) . The centered v ersion of g c is therefore e g c ( v 1 , . . . , v c ) = g c ( v 1 , . . . , v c ) − E h ( V 1 , . . . , V m ) = g c ( v 1 , . . . , v c ) − p ( m, k ) . Lemma 6.5 ( c = 1) . We have g 1 ( v 1 ) = p ( m, k ) and e g 1 ( v 1 ) = 0 for every v 1 ∈ S k − 1 . Henc e ζ 1 = 0 . Pr o of. Indeed, by rotation inv ariance, for every v 1 ∈ S k − 1 w e hav e g 1 ( v 1 ) = P [p os( v 1 , V 2 , . . . , V m ) = R k ] = P [p os( V 1 , V 2 , . . . , V m ) = R k ] = p ( m, k ) . Hence e g 1 ( v 1 ) = g 1 ( v 1 ) − p ( m, k ) = 0 and ζ 1 = V ar e g 1 ( V 1 ) = 0. □ The next lemma pro vides an explicit formula for the conditional expectation kernel g c ( v 1 , . . . , v c ) of arbitrary order. The formula is expressed in terms of conic in trinsic volumes. T o eac h closed con vex cone C ⊆ R k one can asso ciate conic intrinsic volumes υ 0 ( C ) , . . . , υ k ( C ), whic h are nonneg- ativ e and satisfy υ 0 ( C ) + · · · + υ k ( C ) = 1. W e refer to [1, 2] and [44, Section 2.3] for their deﬁnition and prop erties. RANDOM POL YHEDRAL CONES 21 Lemma 6.6 (Conditional exp ectation kernels) . L et c ∈ { 1 , . . . , m − 1 } . Supp ose that v 1 , . . . , v c ∈ S k − 1 , k ≥ 2 , ar e such that the p olyhe dr al c one C = p os( v 1 , . . . , v c ) ⊆ R k is not a line ar subsp ac e. Then g c ( v 1 , . . . , v c ) = 2 X i =1 , 3 , 5 ,... i ≤ k k − i X j =0 υ k − j ( C ) 1 { i + j  = k } 2 m − c  m − c i + j  + 1 { i + j = k } 2 m − c m − c X ℓ = k  m − c ℓ  ! , (6.11) wher e υ 0 ( C ) , . . . , υ k ( C ) ar e the c onic intrinsic volumes of the c one C . Pr o of. Put D := p os( V c +1 , . . . , V m ) ⊆ R k . Passing to dual cones, we get g c ( v 1 , . . . , v c ) = P [p os( C ∪ D ) = R k ] = P [ C ◦ ∩ D ◦ = { 0 } ] . Let O b e a random k × k -matrix that is uniformly (Haar) distributed on the group S O ( k ). Addi- tionally , let O b e indep enden t of the σ -algebra V generated b y V 1 , . . . , V n . Since the la w of D (and hence D ◦ ) is inv ariant under S O ( k ), the randomly rotated cone O D ◦ has the same distribution as D ◦ and we can write g c ( v 1 , . . . , v c ) = P [ C ◦ ∩ O D ◦ = { 0 } ] = E h P [ C ◦ ∩ O D ◦ = { 0 } | V ] i . In the conditional probabilit y , the only random elemen t is the random matrix O . T o compute the conditional probability , w e inv oke the conic kinematic form ula whose pro of can b e found in [1, Corollary 5.2] or in [44, Theorem 4.3.5]; for applications see also [1]. The form ula states that for (deterministic) p olyhedral cones A, B ⊆ R k that are not b oth linear subspaces, P [ A ∩ O B = { 0 } ] = 2 X i =1 , 3 , 5 ,... i ≤ k υ k − i ( A ⊕ B ) = 2 X i =1 , 3 , 5 ,... i ≤ k k − i X j =0 υ j ( A ) υ k − i − j ( B ) . Here, A ⊕ B := { ( a, b ) : a ∈ A, b ∈ B } ⊆ R k ⊕ R k denotes the direct (orthogonal) sum of A and B . Recall that v 1 , . . . , v c ∈ S k − 1 are such that C = p os( v 1 , . . . , v c ) (and hence C ◦ ) is not a linear subspace. The kinematic formula giv es P [ C ◦ ∩ O D ◦ = { 0 } | V ] = 2 X i =1 , 3 , 5 ,... i ≤ k k − i X j =0 υ j ( C ◦ ) υ k − i − j ( D ◦ ) = 2 X i =1 , 3 , 5 ,... i ≤ k k − i X j =0 υ k − j ( C ) υ i + j ( D ) , where w e used the duality relation υ j ( C ◦ ) = υ k − j ( C ) and similarly for D . Recall that C is deterministic and D is random. T aking exp ectation gives g c ( v 1 , . . . , v c ) = 2 X i =1 , 3 , 5 ,... i ≤ k k − i X j =0 υ k − j ( C ) E υ i + j ( D ) . (6.12) Next recall that D = p os( V c +1 , . . . , V m ) has the same la w as the random p olyhedral cone W m − c,k . T o pro ceed, we need formulas for the exp ected conic intrinsic v olumes of this cone. Prop osition 6.7 (Expecte d conic in trinsic volumes of W n,d ) . F or al l d ∈ N , n ∈ N we have E [ υ k ( W n,d )] = E h υ k ( W n,d ) 1 {W n,d  = R d } i = 1 2 n  n k  , k ∈ { 0 , . . . , d − 1 } , (6.13) 22 ZAKHAR KABLUCHKO E [ υ d ( W n,d )] = 1 − 1 2 n d − 1 X ℓ =0  n ℓ  = 1 2 n n X ℓ = d  n ℓ  . (6.14) In the sp e cial c ase wher e d ∈ N and n ∈ { 1 , . . . , d } , these e quations state that E [ υ k ( W n,d )] = 1 2 n  n k  , k ∈ { 0 , . . . , n } . (6.15) Pr o of. Equations (6.13) and (6.14) are prov ed in [15, Lemma 5.1] (building on [20, Corollary 4.3]); where it is assumed that n ≥ d . See also [24, Theorem 5.18] for an alternativ e approach. W e sk etch a proof of (6.15) for n ∈ { 1 , . . . , d } ; for background on conic in trinsic volumes and hyper- plane arrangements we refer to [44, Section 5.2]. F or each sign v ector ε = ( ε 1 , . . . , ε n ) ∈ {± 1 } n , consider the random cone C ε := p os( ε 1 U 1 , . . . , ε n U n ). By symmetry , E υ k ( C ε ) = E υ k ( W n,d ) for ev ery ε . On the other hand, with probabilit y 1, the cones C ε are precisely the cham b ers of a h yp erplane arrangemen t in lin( U 1 , . . . , U n ) that has the same in tersection p oset as the co ordinate h yp erplane arrangement in R n , whose c hambers are the 2 n orthan ts. Since the k th conic intrinsic v olume of an orthan t in R n equals 1 2 n  n k  , the Kliv ans–Swartz formula [44, Theorem 5.1.4] yields P ε ∈{± 1 } n υ k ( C ε ) =  n k  a.s. T aking exp ectations gives 2 n E υ k ( W n,d ) = P ε ∈{± 1 } n E υ k ( C ε ) =  n k  , whic h is precisely (6.15). □ Returning to (6.12) and plugging the v alues of E υ i + j ( D ) = E υ i + j ( W m − c,k ) given b y (6.13) an (6.14) yields g c ( v 1 , . . . , v c ) = 2 X i =1 , 3 , 5 ,... i ≤ k k − i X j =0 υ k − j ( C ) 1 { i + j  = k } 2 m − c  m − c i + j  + 1 { i + j = k } 2 m − c m − c X ℓ = k  m − c ℓ  ! . The pro of of Lemma 6.6 is complete. □ W e hav e already seen that g 0 = g 1 ( v 1 ) = p ( m, k ) and hence e g 1 ( v 1 ) = 0 for all v 1 ∈ S k − 1 . The next step is to compute g 2 ( v 1 , v 2 ). Lemma 6.8 (Conditional exp ectation k ernel for c = 2) . L et v 1 , v 2 ∈ S k − 1 , k ≥ 2 , b e two ve ctors such that v 1  = ± v 2 . Then, e g 2 ( v 1 , v 2 ) = 1 2 m − 1  m − 2 k − 1   2 π arccos ⟨ v 1 , v 2 ⟩ − 1  = − 1 2 m − 1  m − 2 k − 1  · 2 π arcsin ⟨ v 1 , v 2 ⟩ . Pr o of. First w e compute g 2 ( v 1 , v 2 ) by applying Lemma 6.6 with c = 2. The cone C = p os( v 1 , v 2 ) is a w edge with angle α = arccos ⟨ v 1 , v 2 ⟩ (measured in radians). Its conic intrinsic volumes are given b y υ 0 ( C ) = 1 2 − α 2 π , υ 1 ( C ) = 1 2 , υ 2 ( C ) = α 2 π , and υ m ( C ) = 0 whenev er m ≥ 3. So υ k − j ( C ) = 0 unless k − j ∈ { 0 , 1 , 2 } . It follo ws that only the pairs ( i, j ) = (1 , k − 1) and ( i, j ) = (1 , k − 2) con tribute to the sum in (6.11). So g 2 ( v 1 , v 2 ) = 2 υ 1 ( C ) · 1 2 m − 2 m − 2 X ℓ = k  m − 2 ℓ  + 2 υ 2 ( C ) · 1 2 m − 2  m − 2 k − 1  = 1 2 m − 2 m − 2 X ℓ = k  m − 2 ℓ  + α 2 m − 2 π  m − 2 k − 1  . RANDOM POL YHEDRAL CONES 23 W endel’s form ula (1.1) and the deﬁning prop erty of the P as cal triangle giv e p ( m, k ) = P [ W m,k = R k ] = 1 2 m − 1 m − 1 X ℓ = k  m − 1 ℓ  = 1 2 m − 2 m − 2 X ℓ = k  m − 2 ℓ  + 1 2 m − 1  m − 2 k − 1  . Plugging these form ulas in to e g 2 ( v 1 , v 2 ) = g 2 ( v 1 , v 2 ) − p ( m, k ) gives the stated form ula for e g 2 ( v 1 , v 2 ). □ 6.5. Pro of of Theorem 6.1: The v ariance. W e hav e already seen that ζ 1 = V ar g 1 ( V 1 ) = 0. In the presen t section, we compute ζ 2 = V ar g 2 ( V 1 , V 2 ). W e shall see that ζ 2 > 0, whic h means that the U -statistic U n is degenerate of order 2. Lemma 6.9 (Angle b etw een t wo random unit vectors) . L et V 1 , V 2 b e indep endent and uniformly distribute d on S k − 1 , k ≥ 2 . L et Θ := arccos ⟨ V 1 , V 2 ⟩ ∈ [0 , π ] b e the angle b etwe en these ve ctors. Then, E Θ = π 2 , V ar Θ = 1 2 ψ 1  k 2  , wher e ψ 1 ( x ) = d 2 d x 2 log Γ( x ) = P ∞ j =0 1 ( j + x ) 2 > 0 , with x > 0 , is the trigamma function. Pr o of. It is a standard fact, see, e.g., [26, Example 2.28], that Θ has densit y f Θ ( θ ) = (sin θ ) k − 2 Z k , θ ∈ [0 , π ] , Z k = Z π 0 (sin u ) k − 2 d u = √ π Γ  k − 1 2  Γ  k 2  , (6.16) for k ≥ 2. It follows that Θ has the same la w as π − Θ, hence E Θ = π / 2. Next, it follows that V ar(Θ) = E   Θ − π 2  2  = R π 0  θ − π 2  2 (sin θ ) k − 2 d θ R π 0 (sin θ ) k − 2 d θ = R π / 2 − π / 2 x 2 cos k − 2 x d x R π / 2 − π / 2 cos k − 2 x d x , (6.17) where w e used the substitution x = θ − π / 2. It remains to show that the quotient on the righ t-hand side equals 1 2 ψ 1 ( k / 2). W e use the identit y (see [16, Eq. 3.631.9]) F ( a ) := Z π / 2 − π / 2 cos k − 2 x cos( ax ) d x = 2 π Γ( k − 1) 2 k − 1 Γ  k 2 + a 2  Γ  k 2 − a 2  , k > 1 , a ∈ C . (6.18) On the one hand, the quotien t in (6.17) equals − F ′′ (0) /F (0). On the other hand, taking logarithms in (6.18), diﬀerentiating twice, and setting a = 0 gives (log F ) ′ (0) = 0 and (log F ) ′′ (0) = − 1 2 ψ 1 ( k / 2). Hence, − F ′′ (0) /F (0) = − (log F ) ′′ (0) = 1 2 ψ 1 ( k / 2). □ Corollary 6.10 (F ormula for ζ 2 ) . F or k ≥ 2 we have ζ 2 = 2 2 2 m − 2 π 2  m − 2 k − 1  2 · ψ 1  k 2  > 0 . Pr o of. Recall that ζ 2 = V ar g 2 ( V 1 , V 2 ) = V ar e g 2 ( V 1 , V 2 ). Hence ζ 2 = V ar e g 2 ( V 1 , V 2 ) = 1 2 2 m − 2  m − 2 k − 1  2 · 4 π 2 V ar arccos ⟨ V 1 , V 2 ⟩ = 2 2 2 m − 2 π 2  m − 2 k − 1  2 · ψ 1  k 2  , using ﬁrst Lemma 6.8 and then Lemma 6.9. □ W e are now able to pro ve the asymptotic formula for the v ariance stated in (6.4). 24 ZAKHAR KABLUCHKO Prop osition 6.11 (Asymptotics of the v ariance) . L et k ≥ 2 and q ∈ N b e ﬁxe d. Then V ar f d − q ( W d + k,d )  d + k d − q  ! ∼  k + q 2  2  k + q − 2 k − 1  2 ψ 1  k 2  2 2 k +2 q − 4 π 2 · d 2 , d → ∞ . Pr o of. W e ha ve shown that ζ 1 = 0 and ζ 2 > 0. Equation (6.8) with c = 2 gives V ar f d − q ( W d + k,d )  d + k d − q  ! = V ar U n ∼  m 2  2 2! n 2 ζ 2 =  m 2  2  m − 2 k − 1  2 ψ 1  k 2  2 2 m − 4 π 2 · n 2 , n → ∞ . Recalling that m = k + q is ﬁxed and n = d + k ∼ d gives the stated asymptotic equiv alence. □ 6.6. Pro of of Theorem 6.1: Conv olution op erator on the sphere and its eigenv alues. Let k ≥ 2 b e integer and recall that ν k is the uniform probability measure on the unit sphere S k − 1 = { v ∈ R k : ∥ v ∥ = 1 } . Deﬁne the integral op erator A : L 2 ( S k − 1 , ν k ) → L 2 ( S k − 1 , ν k ) by ( Af )( v ) = Z S k − 1 2 π arcsin  ⟨ v , u ⟩  f ( u ) ν k (d u ) , f ∈ L 2 ( S k − 1 , ν k ) , v ∈ S k − 1 . (6.19) Note that A is Hilb ert–Sc hmidt (hence compact) since its kernel K ( u, v ) = 2 π arcsin( ⟨ v , u ⟩ ) is square integrable w.r.t. ν k ⊗ ν k . Also, A is self-adjoint since K is real-v alued and symmetric in its argumen ts. By the sp ectral theorem for compact op erators, A has discrete sp ectrum consisting of coun tably many eigen v alues. Our aim is to describ e the full sp ectral dec omp osition of A (i.e. its eigenv alues and the cor- resp onding eigenspaces). This will b e done in terms of spherical harmonics. Let us recall some basic facts from the harmonic analysis on the sphere; see [10, Chapters 1,2], [3, Chapter 5], [38]. F or n ∈ N 0 let H n ⊆ L 2 ( S k − 1 , ν k ) b e the space of spherical harmonics of degree n . One p ossible deﬁnition of H n is this one: H n = n Y ∈ C ∞ ( S k − 1 ) : ∆ S k − 1 Y = − n ( n + k − 2) Y o , where ∆ S k − 1 denotes the Laplace–Beltrami op erator on S k − 1 . It is known [3, Prop osition 5.8 on p. 78] that the linear space H n is ﬁnite-dimensional with d n,k := dim H n =  n + k − 1 n  −  n + k − 3 n − 2  . Moreo ver, H m ⊥ H n for m  = n , and the linear span of S n ≥ 0 H n is dense in L 2 ( S k − 1 , ν k ); see [10, Theorem 2.2.2]. Note that the excluded case k = 1 is diﬀerent since then the Hilb ert space L 2 ( S k − 1 , ν k ) b ecomes 2-dimensional — see Section 6.8 for this case. Lemma 6.12 (Sp ectral decomp osition of A ) . F or every n ∈ N 0 and every de gr e e n spheric al harmonic Y ∈ H n , we have AY = λ n Y , wher e λ n = 0 ( if n is even ) , λ n = Γ  k 2  Γ  n 2  π Γ  n + k 2  ! 2 =  B  n 2 , k 2  2 π 2 ( if n is o dd ) . T o pro ve Lemma 6.12, we represen t A as the square of another op erator. RANDOM POL YHEDRAL CONES 25 Lemma 6.13 (Square ro ot of A ) . We have A = S 2 , wher e S : L 2 ( S k − 1 , ν k ) → L 2 ( S k − 1 , ν k ) is a line ar op er ator deﬁne d by ( S f )( v ) = Z S k − 1 sign  ⟨ v , u ⟩  f ( u ) ν k (d u ) , f ∈ L 2 ( S k − 1 , ν k ) , v ∈ S k − 1 . Pr o of. Let U b e uniformly distributed on S k − 1 . F or all v , w ∈ S k − 1 it is easy to c heck (see, e.g., [51, Lemma 6.7 on p. 144]) that E  sign  ⟨ v , U ⟩  sign  ⟨ U, w ⟩  = 2 π arcsin( ⟨ v , w ⟩ ) . Applying the deﬁnition of S twice gives ( S 2 f )( v ) = Z S k − 1  Z S k − 1 sign  ⟨ v , u ⟩  sign  ⟨ u, w ⟩  ν k (d u )  f ( w ) ν k (d w ) . Th us S 2 is an in tegral op erator with kernel L ( v , w ) := Z S k − 1 sign  ⟨ v , u ⟩  sign  ⟨ u, w ⟩  ν k (d u ) = E  sign  ⟨ v , U ⟩  sign  ⟨ U, w ⟩  = 2 π arcsin( ⟨ v , w ⟩ ) . W e conclude that A = S 2 . □ The next lemma describ es the sp ectral decomp osition of S and, together with the formula A = S 2 , implies Lemma 6.12. Lemma 6.14 (Sp ectral decomp osition of S ) . F or every n ∈ N 0 and every Y ∈ H n , we have S Y = τ n Y (i.e. the r estriction of S to H n is the multiplic ation by τ n ), wher e τ n = 0 ( if n is even ) , τ n = ( − 1) n − 1 2 Γ  k 2  Γ  n 2  π Γ  n + k 2  , ( if n is o dd ) . Pr o of. The main to ol in this pro of is the F unk–Heck e formula [10, Theorem 1.2.9] which we no w recall. Let α = k − 2 2 ≥ 0 and let h : [ − 1 , 1] → C b e in tegrable with resp ect to the measure (1 − t 2 ) α − 1 2 d t . Deﬁne the integral operator T h : L 2 ( S k − 1 , ν k ) → L 2 ( S k − 1 , ν k ) by ( T h f )( v ) := Z S k − 1 h ( ⟨ v , u ⟩ ) f ( u ) ν k (d u ) . The F unk–Heck e form ula states that for every n ∈ N 0 and ev ery degree n spherical harmonic Y ∈ H n , we ha ve T h Y = λ n ( h ) Y , where λ n ( h ) = b α C ( α ) n (1) Z 1 − 1 h ( t ) C ( α ) n ( t ) (1 − t 2 ) α − 1 2 d t, b α =  Z 1 − 1 (1 − t 2 ) α − 1 2 d t  − 1 = Γ( α + 1) √ π Γ( α + 1 2 ) . Here C ( α ) n denotes the Gegenbauer p olynomial [10, Sections B.1,B.2] of degree n . F or our purp oses, it is con venien t to deﬁne it by the Ro drigues’ formula (see [10, Equations (B.1.2), (B.2.1)]) C ( α ) n ( t ) (1 − t 2 ) α − 1 2 = κ ( α ) n d n d t n  (1 − t 2 ) n + α − 1 2  , κ ( α ) n = ( − 1) n 2 n n ! Γ  α + 1 2  Γ( n + 2 α ) Γ(2 α )Γ  n + α + 1 2  . In our setting, h ( t ) = sign( t ), and the F unk–Heck e form ula giv es S Y = τ n Y for all Y ∈ H n , where τ n = b α C ( α ) n (1) I ( α ) n with I ( α ) n := Z 1 − 1 sign( t ) C ( α ) n ( t ) (1 − t 2 ) α − 1 2 d t. 26 ZAKHAR KABLUCHKO It remains to compute I ( α ) n . W riting F n ( t ) := (1 − t 2 ) n + α − 1 2 , splitting the in tegral and using the fact that the antideriv ativ e of F ( n ) n is F ( n − 1) n giv es I ( α ) n = κ ( α ) n Z 1 − 1 sign( t ) F ( n ) n ( t ) d t = − κ ( α ) n F ( n − 1) n ( t )   t =0 t = − 1 + κ ( α ) n F ( n − 1) n ( t )   t =1 t =0 . Since the v anishing order of F ( t ) at t = ± 1 is n + α − 1 2 > n − 1, w e hav e F ( n − 1) n ( ± 1) = 0. It follo ws that I ( α ) n = − 2 κ ( α ) n F ( n − 1) n (0) . T aking these results together gives τ n = b α C ( α ) n (1) I ( α ) n = b α C ( α ) n (1) · ( − 2 κ ( α ) n ) F ( n − 1) n (0) = ( − 1) n +1 2 n − 1 Γ( α + 1) √ π F ( n − 1) n (0) Γ( n + α + 1 2 ) , (6.20) where we used that κ ( α ) n = ( − 1) n 2 n n ! Γ  α + 1 2  Γ( n + 2 α ) Γ(2 α )Γ  n + α + 1 2  , b α = Γ( α + 1) √ π Γ  α + 1 2  , C ( α ) n (1) = Γ( n + 2 α ) Γ(2 α ) n ! ; see [10, Equation (B.2.2)] for the form ula for C ( α ) n (1). The function F ( t ) is ev en and (6.20) implies that τ n = 0 for ev en n . Let n b e o dd. Expanding around t = 0 gives F n ( t ) = (1 − t 2 ) n + α − 1 2 = ∞ X j =0 ( − 1) j  n + α − 1 2 j  t 2 j . Then n − 1 is ev en and the term contributing to t n − 1 is j = n − 1 2 , hence F ( n − 1) n (0) = ( n − 1)! ( − 1) n − 1 2  n + α − 1 2 n − 1 2  = ( − 1) n − 1 2 Γ( n ) Γ  n + α + 1 2  Γ  n +1 2  Γ  α + n +2 2  . Plugging this in to (6.20) and using Legendre’s duplication form ula Γ( n ) = 2 n − 1 Γ ( n 2 ) Γ ( n +1 2 ) √ π giv es τ n = ( − 1) n − 1 2 Γ( α + 1) Γ  n 2  π Γ  α + n 2 + 1  = ( − 1) n − 1 2 Γ  k 2  Γ  n 2  π Γ  n + k 2  , n o dd . In the last step, w e used α = k − 2 2 . □ Prop osition 6.15 (T races of A and A 2 ) . The op er ator A deﬁne d in (6.19) is of tr ac e class and tr( A ) = X n =1 , 3 , 5 ,... d n,k Γ  k 2  Γ  n 2  π Γ  n + k 2  ! 2 = 1 , tr( A 2 ) = X n =1 , 3 , 5 ,... d n,k Γ  k 2  Γ  n 2  π Γ  n + k 2  ! 4 = 2 π 2 ψ 1  k 2  , wher e d n,k = dim H n =  n + k − 1 n  −  n + k − 3 n − 2  and ψ 1 is the trigamma function. RANDOM POL YHEDRAL CONES 27 Pr o of. Let us ﬁrst sho w that A, A 2 are trace class op erators and tr( A ) = 1, tr( A 2 ) = 2 π 2 ψ 1 ( k 2 ). Recall that A = S 2 = S S ∗ and S = S ∗ is Hilbert–Schmidt. It follows that A is trace class and tr( A ) = ∥ S ∥ 2 HS is the squared Hilb ert–Schmidt norm of S . Since the kernel of S takes only v alues ± 1, we ha ve ∥ S ∥ 2 HS = 1 and hence tr( A ) = 1. Similarly , A 2 = AA ∗ and A = A ∗ is Hilb ert–Schmidt, hence A 2 is of trace class and tr( A 2 ) = ∥ A ∥ 2 HS = 4 π 2 Z Z S k − 1 × S k − 1 arcsin 2  ⟨ v , u ⟩  ν k (d v ) ν k (d u ) = 4 π 2 Z S k − 1 arcsin 2  ⟨ v , e 1 ⟩  ν k (d v ) = 4 π 2 Γ  k 2  √ π Γ  k − 1 2  Z 1 − 1 arcsin 2 ( t ) (1 − t 2 ) k − 3 2 d t = 4 π 2 Γ  k 2  √ π Γ  k − 1 2  Z π / 2 − π / 2 x 2 cos k − 2 x d x, where the last step follows b y the substitution t = sin x . W e ha ve already computed the righ t-hand side in the pro of of Lemma 6.9: it equals 4 π 2 V ar Θ = 2 π 2 ψ 1 ( k 2 ). So, tr( A 2 ) = 2 π 2 ψ 1 ( k 2 ). On the other hand, w e hav e seen in Lemma 6.12 that the nonzero eigenv alues of A are λ n = Γ  k 2  Γ  n 2  π Γ  n + k 2  ! 2 (with multiplicit y d n,k = dim H n ) , n = 1 , 3 , 5 , . . . . Note that 0 is also an eigenv alue of A , and the eigenspace of 0 is the closure of ⊕ n =0 , 2 , 4 ,... H n , ho wev er, 0 do es not contribute to the trace. It follows that tr( A ) = P n =1 , 3 , 5 ,... λ n d n,k and tr( A 2 ) = P n =1 , 3 , 5 ,... λ 2 n d n,k , which completes the pro of. □ Example 6.16 ( k = 2) . F or k = 2 the underlying space is the unit circle S 1 . W e represen t v ∈ S 1 as v = (cos θ , sin θ ). The spaces of spherical harmonics are as follows: H 0 = lin { 1 } and H n = lin { cos( nθ ) , sin( nθ ) } for n ≥ 1. So dim H 0 = 1 and dim H n = 2 for n ≥ 1. The non-zero eigen v alues of the op erator A are 4 / ( π 2 n 2 ), n = 1 , 3 , 5 , . . . , each with m ultiplicity 2. This can b e v eriﬁed by direct computation. 6.7. Pro of of Theorem 6.1: Completing the argument. W e now gathered all ingredien ts needed to pro ve Theorem 6.1 for k ≥ 2. Recall from (6.10) the representation f d − q ( W d + k,d ) /  d + k d − q  = U n , where U n is the U -statistic corresp onding to the kernel h ( v 1 , . . . , v m ) = 1 { pos( v 1 ,...,v m )= R k } , for all v 1 , . . . , v m ∈ S k − 1 . In Lemma 6.5 w e ha ve shown that ζ 1 = 0 and in Lemma 6.8 we computed e g 2 ( v 1 , v 2 ) = − 1 2 m − 1  m − 2 k − 1  · 2 π arcsin ⟨ v 1 , v 2 ⟩ , v 1  = ± v 2 . In Corollary 6.10 we ha ve seen that ζ 2 > 0. W e are th us in the setting of Theorem 6.4. The op erator T : L 2 ( S k − 1 , ν k ) → L 2 ( S k − 1 , ν k ) app earing in Theorem 6.4 is ( T f )( x ) := Z S k − 1 e g 2 ( x, y ) f ( y ) ν k (d y ) = − 1 2 m − 1  m − 2 k − 1  · ( Af )( x ) , where A is the op erator we analyzed in Section 6.6. W e hav e seen in Lemma 6.12 that the non-zero eigen v alues of the op erator A are λ r = Γ  k 2  Γ  r 2  π Γ  r + k 2  ! 2 with multiplicit y d r,k =  r + k − 1 r  −  r + k − 3 r − 2  , r = 1 , 3 , 5 , . . . . 28 ZAKHAR KABLUCHKO The non-zero eigenv alues of T are then − 1 2 m − 1  m − 2 k − 1  λ r with r = 1 , 3 , 5 , . . . and the same multiplic- ities d r,k as b efore. No w w e hav e everything to apply Theorem 6.4. Theorem 6.4 gives n ( U n − EU n ) w − → n →∞ −  m 2  m − 2 k − 1  2 m − 1 X r =1 , 3 , 5 ,... d r,k X j =1 λ r ( ξ 2 r,j − 1) , where ξ r,j are i.i.d. standard normal random v ariables indexed by r ∈ { 1 , 3 , 5 , . . . } and j ∈ { 1 , . . . , d r,k } . In Lemma 6.15 we hav e seen that tr( A ) = P ∞ r =1 , 3 , 5 ,... d r,k λ r = 1, whic h allo ws us to rewrite the ab ov e result as n ( U n − EU n ) w − → n →∞  m 2  m − 2 k − 1  2 m − 1 (1 − Q k ) with Q k := X r =1 , 3 , 5 ,... λ r d r,k X j =1 ξ 2 r,j . No w observe that P d r,k j =1 ξ 2 r,j is distributed as Gamma( d r,k 2 , 1 2 ) (chi squared distribution) and these v ariables are indep endent for diﬀerent r ’s. This prov es the distributional represen tation for Q k stated in Theorem 6.1. By Lemma 6.15, E Q k = tr( A ) = 1. Similarly , the v ariance of Q k is given b y V ar Q k = X r =1 , 3 , 5 ,... λ 2 r d r,k X j =1 V ar( ξ 2 r,j ) = tr( A 2 ) V ar( ξ 2 1 , 1 ) = 2 π 2 ψ 1  k 2  V ar( ξ 2 1 ) = 4 π 2 ψ 1  k 2  , where the formula for tr( A 2 ) comes from Lemma 6.15. The asymptotic formula for V ar U n stated in (6.4) has b een already established in Proposition 6.11. The pro of of Theorem 6.1 in the case where k ≥ 2 is complete. 6.8. Pro of of Theorem 6.1: Case k = 1 . The case where k = 1 requires sp ecial treatment since the unit sphere b ecomes degenerate, S 0 = {± 1 } , and the parts of the pro of based on the conic kinematic formula and spherical harmonics do not apply directly . Our setting is as follo ws: • The sample space is X = S 0 = {± 1 } , endow ed with the uniform distribution ν 1 . • V 1 , V 2 , . . . , V n , where n = d + 1, are i.i.d. with P [ V i = 1] = P [ V i = − 1] = 1 / 2. • The order of the U -statistic is m := n − ℓ = q + 1 and the kernel h : {± 1 } m → { 0 , 1 } is h ( v 1 , . . . , v m ) := 1 { pos( v 1 ,...,v m )= R } = 1 − 1 { v 1 = ... = v m } . The asso ciated U -statistic is U n = f d − q ( W d +1 ,d ) /  d +1 d − q  ; see (6.10). Lemma 6.17 (Conditional exp ectation kernels for k = 1) . We have g 0 = EU n = 1 − 2 1 − m , e g 1 ( x ) = 0 for x ∈ {± 1 } and e g 2 ( x, y ) = ( +2 1 − m , x  = y , − 2 1 − m , x = y , x, y ∈ {± 1 } . (6.21) Conse quently, ζ 1 = 0 and ζ 2 = 2 2 − 2 m . Pr o of. Clearly , g 0 = EU n = E h ( V 1 , . . . , V m ) = 1 − P [ V 1 = . . . = V m ] = 1 − 2 · 2 − m . RANDOM POL YHEDRAL CONES 29 Next, for x ∈ {± 1 } , g 1 ( x ) = E  h ( x, V 2 , . . . , V m )  = 1 − P  x, V 2 , . . . , V m are all equal  = 1 − 2 − ( m − 1) = g 0 . Hence e g 1 ( x ) = g 1 ( x ) − g 0 = 0 and ζ 1 = 0. Next, for x, y ∈ {± 1 } , g 2 ( x, y ) = E  h ( x, y , V 3 , . . . , V m )  = 1 − P  x, y , V 3 , . . . , V m are all equal  . If x  = y , then the ev ent “all equal” is impossible and hence g 2 ( x, y ) = 1. If x = y , then “all equal” o ccurs iﬀ V 3 = · · · = V m = x , which has probability 2 − ( m − 2) . Therefore g 2 ( x, y ) =    1 , x  = y , 1 − 2 − ( m − 2) , x = y , x, y ∈ {± 1 } . Hence the centered k ernel e g 2 ( x, y ) = g 2 ( x, y ) − g 0 = g 2 ( x, y ) − 1 + 2 · 2 − m is given b y (6.21). Finally , ζ 2 = E [ e g 2 ( V 1 , V 2 ) 2 ] = (2 1 − m ) 2 = 2 2 − 2 m . □ As a linear space, L 2 ( {± 1 } , ν 1 ) can b e iden tiﬁed with R 2 b y identifying a function f with the v ector ( f (+1) , f ( − 1)) ⊤ . The in tegral op erator T : L 2 ( {± 1 } , ν 1 ) → L 2 ( {± 1 } , ν 1 ) with kernel e g 2 ( x, y ) app earing in Theorem 6.4 is then represented by the matrix T ≡ 2 − m  − 1 1 1 − 1  . In particular, T has eigen v alues 0 and − 2 1 − m . Recall that n = d + 1 ∼ d and m = q + 1. Theorem 6.4 yields n · f d − q ( W d +1 ,d ) − E f d − q ( W d +1 ,d )  d +1 d − q  ! = n ( U n − EU n ) w − → n →∞ −  m 2  2 1 − m ( ξ 2 − 1) =  q + 1 2  2 − q (1 − ξ 2 ) , where ξ is standard normal and hence Q 1 := ξ 2 d = Gamma( 1 2 , 1 2 ). Applying (6.8) and recalling that ζ 2 = 2 − 2 q giv es V ar f d − q ( W d +1 ,d )  d +1 d − q  ! = V ar U n ∼ 2  m 2  2 n 2 ζ 2 ∼  q +1 2  2 d 2 2 1 − 2 q . Note ﬁnally that E [ ξ 2 ] = 1 and V ar[ ξ 2 ] = 2 = 4 π 2 ψ 1 ( 1 2 ). This prov es Theorem 6.1 for k = 1. 6.9. Exp onential concen tration of face coun ts. Let us ﬁnally mention the follo wing non- asymptotic concentration result. Recall that p ( n, k ) is given b y W endel’s formula (1.1). Theorem 6.18 (Exp onen tial concen tration of face counts) . L et d ∈ N , n > d , and ℓ ∈ { 0 , . . . , d − 1 } . Then, for al l t > 0 , P "      f ℓ ( W n,d )  n ℓ  − p ( n − ℓ, n − d )      ≥ t # ≤ 2e − 2 t 2 ⌊ n/ ( n − ℓ ) ⌋ . F or the pro of we need a result of Ho eﬀding [19] which can b e found in the b o ok of Serﬂing [47, Theorem A on p. 201] and in [41]. 30 ZAKHAR KABLUCHKO Lemma 6.19 (Ho eﬀding b ound for U -statistics) . L et the kernel h : X m → R b e b ounde d, that is L := sup x ∈X m h ( x ) − inf x ∈X m h ( x ) < ∞ . Then for every n ≥ m and t > 0 , we have P [ | U n − EU n | ≥ t ] ≤ 2e − 2 ⌊ n/m ⌋ t 2 /L 2 . Pr o of of The or em 6.18. Recall from Section 6.3 (in particular, Equation (6.9)) the identiﬁcation U n =  n n − ℓ  − 1 X 1 ≤ i 1 < ··· 0 , . . . , λ m > 0 such that λ 1 v 1 + · · · + λ m v m = 0 . Pr o of. (i) ⇒ (ii). Assume (i). Then for eac h k ∈ [ m ] we ha ve − v k ∈ lin( v 1 , . . . , v m ) = p os( v 1 , . . . , v m ), so there exist co eﬃcients µ 1; k ≥ 0 , . . . , µ m ; k ≥ 0 with − v k = m X j =1 µ j ; k v j . RANDOM POL YHEDRAL CONES 31 Summing these identities o ver k = 1 , . . . , m and rearranging yields P m j =1 λ j v j = 0 with λ j := 1 + P m k =1 µ j ; k ≥ 1. Hence λ 1 > 0 , . . . , λ m > 0, proving (ii). (ii) ⇒ (i). Assume P m j =1 λ j v j = 0 with all λ j > 0. Then for ev ery k ∈ [ m ] w e hav e − v k = X j : j  = k ( λ j /λ k ) v j ∈ p os( v 1 , . . . , v m ) , implying ± v k ∈ p os( v 1 , . . . , v m ) for every k . Therefore lin( v 1 , . . . , v m ) ⊆ p os( v 1 , . . . , v m ). The rev erse inclusion p os( v 1 , . . . , v m ) ⊆ lin( v 1 , . . . , v m ) alwa ys holds, proving (i). □ The next lemma makes precise how faces of the cone p os( a 1 , . . . , a n ) corresp ond to strictly p ositiv e dep enden t subsets (those for which pos = lin) in a Gale-dual conﬁguration. It can be found in [48, Theorem 1 on p. 167] or [34, 2A11 on p. 95]. Lemma A.2 (F aces vs. p os = lin) . L et d ′ , d ′′ ∈ N and put n := d ′ + d ′′ . L et a 1 , . . . , a n ∈ R d ′ and b 1 , . . . , b n ∈ R d ′′ b e in line ar Gale duality, and set C A := p os( a 1 , . . . , a n ) . F or I ⊆ [ n ] , the fol lowing ar e e quivalent: (i) Ther e exists a fac e F of C A such that I = { i ∈ [ n ] : a i ∈ F } . (ii) p os( b j : j ∈ I c ) = lin( b j : j ∈ I c ) . Her e I c := [ n ] \ I is the c omplement of I . Pr o of. Let A ∈ R d ′ × n and B ∈ R d ′′ × n b e the matrices with columns a 1 , . . . , a n and b 1 , . . . , b n , resp ectiv ely . Step 1. Every face of C A is of the form F = { y ∈ C A : ⟨ u, y ⟩ = 0 } for some u ∈ R d ′ suc h that ⟨ u, y ⟩ ≥ 0 for all y ∈ C A . (When F = C A , we ma y tak e u = 0.) Thus, (i) holds if and only if there exists u ∈ R d ′ suc h that ⟨ u, a i ⟩ = 0 (for i ∈ I ) and ⟨ u, a j ⟩ > 0 (for j ∈ I c ) . With x := u ⊤ A ∈ R n , (i) is equiv alent to the existence of a vector x ∈ Row( A ) with x i = 0 (for i ∈ I ) , x j > 0 (for j ∈ I c ) . (A.2) Step 2. Apply Lemma A.1 to the v ectors b j , j ∈ I c . Then (ii) is equiv alent to the existence of co eﬃcien ts x j > 0 for j ∈ I c suc h that P j ∈ I c x j b j = 0. Extending b y x i := 0 for i ∈ I and deﬁning x = ( x 1 , . . . , x n ), we see that (ii) is equiv alent to the existence of a v ector x ∈ Row( B ) ⊥ with x i = 0 (for i ∈ I ) , x j > 0 (for j ∈ I c ) . (A.3) Step 3. By (A.1), Row( A ) = Ro w ( B ) ⊥ , proving the equiv alence of (A.2) and (A.3). Hence (i) and (ii) are equiv alent. □ Recall that pairwise distinct vectors v 1 , . . . , v m ∈ R d , where m ≥ d , are in general linear p osition if every d -elemen t subset of { v 1 , . . . , v m } is linearly indep endent. Lemma A.3. L et d ′ , d ′′ ∈ N and put n := d ′ + d ′′ . L et a 1 , . . . , a n ∈ R d ′ and b 1 , . . . , b n ∈ R d ′′ b e in line ar Gale duality. If one of the c onﬁgur ations is in gener al line ar p osition, then the other c onﬁgur ation is in gener al line ar p osition as wel l. 32 ZAKHAR KABLUCHKO Pr o of. Assume that a 1 , . . . , a n are in general linear p osition. Let J ⊆ [ n ] with # J = d ′′ and put I := J c , so # I = d ′ . T o sho w that b j , j ∈ J , are linearly indep endent, let x ∈ R n satisfy x 1 b 1 + . . . + x n b n = 0 and x i = 0 for all i ∈ I (so x is supp orted on J ). Th us x ∈ Ro w ( B ) ⊥ and, b y (A.1), x ∈ Row( A ). Let A I b e the ( d ′ × d ′ )-submatrix of A consisting of the columns indexed b y I . Since # I = d ′ and a 1 , . . . , a n are in general p osition, the matrix A I is in vertible. Now, x ∈ Ro w ( A ) is a linear combination of the ro ws of A and x i = 0 for all i ∈ I . On the other hand, the rows of A I are linearly independent. It follows that x is a trivial linear combination of the ro ws of A and hence x = 0. This shows that the columns b j , j ∈ J , are linearly indep endent. □ Under the additional assumption of general linear p osition, the face-dep endence corresp on- dence of Lemma A.2 simpliﬁes and can b e phrased in terms of positive spanning the whole space on the Gale-dual side. W e are now ready to pro ve Lemma 2.2, which w e restate for con v enience. Lemma A.4 (F aces vs. p ositive spanning subsets under general linear p osition) . L et d ′ , d ′′ ∈ N and put n := d ′ + d ′′ . L et a 1 , . . . , a n ∈ R d ′ and b 1 , . . . , b n ∈ R d ′′ b e in line ar Gale duality, and assume that b oth c onﬁgur ations ar e in gener al line ar p osition. L et I ⊆ [ n ] satisfy # I ≤ d ′ − 1 . Then the fol lowing ar e e quivalent: (i) p os( a i : i ∈ I ) is a fac e of C A := p os( a 1 , . . . , a n ) . (ii) p os( b j : j ∈ I c ) = R d ′′ . Her e I c := [ n ] \ I is the c omplement of I . Pr o of. W e ﬁrst consider the case I = ∅ . Then p os( a i : i ∈ I ) = { 0 } , so (i) sa ys that { 0 } is a face of C A . Recall that a 1 , . . . , a n are in general linear p osition, in particular a i  = 0 for all i ∈ [ n ]. Hence { 0 } is a face of C A if and only if there exists a face F of C A con taining none of the generators a 1 , . . . , a n . Thus w e ma y apply Lemma A.2 with I = ∅ , which giv es that (i) is equiv alent to p os( b 1 , . . . , b n ) = lin( b 1 , . . . , b n ) . By Deﬁnition 2.1, we hav e lin( b 1 , . . . , b n ) = R d ′′ . Th us (i) is equiv alent to p os( b 1 , . . . , b n ) = R d ′′ , whic h is exactly (ii). Hence the lemma holds for I = ∅ . F rom no w on assume 1 ≤ # I ≤ d ′ − 1. Claim. If a 1 , . . . , a n are in general linear p osition and I ⊆ [ n ] satisﬁes 1 ≤ # I ≤ d ′ − 1, then { k ∈ [ n ] : a k ∈ p os( a i : i ∈ I ) } = I . (A.4) Pr o of of the claim. Clearly I is con tained in the set on the left. Con versely , let k ∈ [ n ] be suc h that a k ∈ p os( a i : i ∈ I ). Then a k ∈ lin( a i : i ∈ I ). If k / ∈ I , this gives a non-trivial linear dep endence among the # I + 1 ≤ d ′ v ectors a i , i ∈ I ∪ { k } . Since a 1 , . . . , a n are in general linear position, every set of at most d ′ v ectors is linearly indep endent, a contradiction. This prov es (A.4). □ (i) ⇒ (ii). Assume F := p os( a i : i ∈ I ) is a face of C A . By the claim, I = { i ∈ [ n ] : a i ∈ F } . Hence Lemma A.2 yields p os( b j : j ∈ I c ) = lin( b j : j ∈ I c ) . Since # I ≤ d ′ − 1, w e hav e #( I c ) = n − # I ≥ d ′′ + 1. Since b 1 , . . . , b n are in general linear p osition, lin( b j : j ∈ I c ) = R d ′′ . Therefore p os( b j : j ∈ I c ) = R d ′′ , which is (ii). (ii) ⇒ (i). Assume (ii). Then in particular p os( b j : j ∈ I c ) = lin( b j : j ∈ I c ), so by Lemma A.2 there exists a face F of C A suc h that { i ∈ [ n ] : a i ∈ F } = I . In particular, p os( a i : i ∈ I ) ⊆ F . F or RANDOM POL YHEDRAL CONES 33 the reverse inclusion, let x ∈ F ⊆ C A and write x = P n k =1 λ k a k with λ k ≥ 0. If λ k > 0 for some k , then a k ∈ F (b ecause F is a face of the cone C A ), hence k ∈ I recalling that { i ∈ [ n ] : a i ∈ F } = I . Therefore x is a nonnegative linear combination of a i , i ∈ I , i.e. x ∈ p os( a i : i ∈ I ). This sho ws F ⊆ p os( a i : i ∈ I ). Com bining b oth inclusions yields F = p os( a i : i ∈ I ), so p os( a i : i ∈ I ) is a face of C A , proving (i). □ A cknowledgement This pap er b eneﬁted from many helpful and illuminating discussions with ChatGPT, whic h sharp ened b oth the p ersp ective and presentation; the work would hav e b een imp ossible without these exc hanges. Supported by the German Researc h F oundation under German y’s Excellence Strategy EX C 2044/2 – 390685587, Mathematics M ¨ unster: Dynamics - Ge ometry - Structur e and b y the DFG priority program SPP 2265 R andom Ge ometric Systems . References [1] D. Amelunxen and M. 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Random Polyhedral Cones I: Distributional Results via Gale Duality

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