Small faces in stationary Poisson hyperplane tessellations
aa r X i v : . [ m a t h . M G ] A ug Small faces in stationary Poisson hyperplane tessellations
Rolf Schneider
Abstract
We consider the tessellation induced by a stationary Poisson hyperplane process in d -dimensional Euclidean space. Under a suitable assumption on the directional distribution,and measuring the k -faces of the tessellation by a suitable size functional, we determine alimit distribution for the shape of the typical k -face, under the condition of small size andthis tending to zero. The limit distribution is concentrated on simplices. This extends aresult of Gilles Bonnet. Keywords:
Poisson hyperplane tessellation, typical face, small size, limit distribution,simplex shape
Mathematics Subject Classification:
Primary 60D05, Secondary 52C22
A stationary Poisson hyperplane process in R d gives rise to a tessellation of R d into convexpolytopes. The polytopes in such a Poisson hyperplane tessellation pose a considerablenumber of geometrically interesting questions. First, one may notice that generally the varietyof the appearing cells (the d -dimensional polytopes) is very rich: under mild assumptions onthe directional distribution of the hyperplane process (which are satisfied, for example, inthe isotropic case), the translates of the cells in the tessellation are a.s. dense (with respectto the Hausdorff metric) in the space of convex bodies in R d . Further, with probability one,every combinatorial type of a simple d -polytope appears infinitely often among the cells of thetessellation (see [12]). The latter result can be strengthened: from ‘infinitely often’ to ‘withpositive density’, see [13]. A different type of questions comes up if one asks for average cells,formalized by the notions of the ‘zero cell’ and the ‘typical cell’. A well-known question ofD.G. Kendall concerned an isotropic Poisson line tessellation in the plane and asked whetherthe conditional law for the shape of the zero cell, given its area, converges weakly, if the areatends to infinity, to the degenerate law concentrated at the circular shape. Proofs were givenby Kovalenko [9, 10]. This led to detailed investigations into the relations between large sizeand shape, for zero cells and typical cells, in higher dimensions, for non-isotropic tessellations,and for general (axiomatized) notions of size. We refer to [3, 4, 5, 6]. The extension from cellsto lower-dimensional faces revealed new aspects, since the shape of a large face is influencednot only by its size, but also by its direction; see [7, 8].Small cells, on the other side, have been studied with less intensity, so far. If one looks ata sufficiently large simulation of a stationary, isotropic Poisson line process in the plane, onewill notice that the very small cells are mostly triangles, but, of course, of varying shapes andorientations. In his heuristic approach to Kendall’s problem, Miles [11] also briefly consideredsmall cells. We quote his conclusion: “Thus it may be said that the ‘small’ polygons of P aretriangles of random shape, the corresponding shape distribution being immediately dependenton the conditioning characteristic determining ‘smallness’. This contrasts strongly with the1large’ polygons (. . . ), for which shape distributions are characteristically degenerate, eventhough evidently dependent on the particular conditioning variable.” This concerns theisotropic case. Poisson line processes with only two directions of the lines, where no trianglescan be produced, were investigated by Beermann, Redenbach, and Th¨ale [1]. They showedthat the asymptotic shape of cells with small area is degenerate, whereas the asymptoticshape of cells with small perimeter is indeterminate. The first investigation of small cellsin higher dimensions is due to Bonnet [2]. Under various assumptions on the directionaldistribution of the hyperplane process, he found different asymptotic behavior for small cells,including results on the speed of convergence. If the spherical directional distribution ofthe hyperplane process is absolutely continuous with respect to spherical Lebesgue measure,he established that the distribution of the shape of the typical cell, under the condition ofsmall size and this size tending to zero, converges to a distribution on the space of simplexshapes, which was represented explicitly in terms of the directional distribution and theemployed size measurement. The present note extends this result in two directions. First,the assumption of absolute continuity is weakened: the optimal assumption seems to be thatthe spherical directional distribution assigns measure zero to each great subsphere. Second,and more important, we consider also lower-dimensional faces and extend Bonnet’s result tothe typical k -face of the tessellation, for k ∈ { , . . . , d } . This requires, first, to generalize arepresentation of the distribution of the typical cell (Theorem 10.4.6 in [14], in the isotropiccase), involving the inball center as a center function, to typical k -faces and to more generaldirectional distributions. The result is Theorem 1 in Section 4. Then Theorem 2 in Section5 extends Bonnet’s result to the typical k -face. We fix some notation. We work in R d ( d ≥ d -dimensional real vector space with origin o , scalar product h· , ·i and induced norm k · k . The unit ball of R d is denoted by B d , theunit sphere by S d − , and the Lebesgue measure on R d by λ d . By H we denote the space ofhyperplanes in R d , with its usual topology. For K ⊂ R d , we denote by[ K ] H = { H ∈ H : H ∩ K = ∅} the set of hyperplanes meeting K . Every hyperplane in H has a representation H ( u, τ ) = { x ∈ R d : h x, u i = τ } with u ∈ S d − and τ ∈ R , and we have H ( u, τ ) = H ( u ′ , τ ′ ) if and only if ( u, τ ) = ( ǫu ′ , ǫτ ′ )with ǫ ∈ {− , } . The vectors ± u are the unit normal vectors of the hyperplane H ( u, τ ), andusually one of them is chosen arbitrarily as ‘the’ normal vector of the hyperplane. The set H − ( u, τ ) = { x ∈ R d : h x, u i ≤ τ } is one of the closed halfspaces bounded by H ( u, τ ). If H is a hyperplane and x ∈ R d a pointnot in H , we denote by H − x the closed halfspace bounded by H that contains x .The space K of convex bodies (nonempty, compact, convex subsets) of R d is equippedwith the Hausdorff metric. The subspace P of polytopes is a Borel set in K . Generally for atopological space T , we denote by B ( T ) the σ -algebra of Borel sets in T .In the following, b X will always be a non-degenerate stationary Poisson hyperplane processin R d . We refer to [14], in particular Sections 3.2, 4.4. and 10.3, for the basic notions and2undamental facts about such processes. We denote the underlying probability by P , andexpectation by E . The intensity measure b Θ of b X , defined by b Θ( A ) := E b X ( A ) , A ∈ B ( H ) , has a decomposition Z H f d b Θ = b γ Z S d − Z ∞∞ f ( H ( u, τ )) d τ ϕ (d u ) (1)for every nonnegative, measurable function f on H ; see [14, (4.33)]. Here b γ > intensity of b X and ϕ is an even Borel probability measure on S d − , called the sphericaldirectional distribution of b X .For a convex body K ∈ K , the distribution of the number of hyperplanes of b X hitting K is given by P (cid:16) b X ([ K ] H ) = n (cid:17) = e − b Θ([ K ] H ) b Θ([ K ] H ) n n ! , n ∈ N . It follows from (1) that b Θ([ K ] H ) = b γ Z S d − Z ∞−∞ { H ( u, τ ) ∩ K = ∅} d τ ϕ (d u ) = 2 b γ Φ( K )with Φ( K ) = Z S d − h ( K, u ) ϕ (d u ) , where h ( K, u ) = max {h x, u i : x ∈ K } defines the support function h ( K, · ) of K . (By wealways denote an indicator function.) For obvious reasons, we call Φ the hitting functional of b X . For convenience, we often identify a simple counting measure with its support, so we write H ∈ b X ( ω ) for b X ( ω )( { H } ) = 1. In particular, Campbell’s theorem for b X (and for f as above)is written in the form E X H ∈ b X f ( H ) = Z H f d b Θ . (2)The following notions of general position play a role for hyperplane processes. Definition 1.
A system of hyperplanes in R d is in translationally general position if every k -dimensional plane of R d is contained in at most d − k hyperplanes of the system, for k = 0 , . . . , d − .A system of hyperplanes in R d is in directionally general position if any d of the hyper-planes of the system have linearly independent normal vectors.Translationally general position together with directionally general position is called gen-eral position . While the hyperplanes of b X a.s. have the first property, they have the second propertyonly under an assumption on its spherical directional distribution. Definition 2.
A measure on S d − is called subspace-free if it is zero on each great subsphereof S d − .
3n the proof below, and later, we denote by b X m = the set of all ordererd m -tuples of pairwisedistinct hyperplanes from b X . Lemma 1.
Let b X be nondegenerate, stationary Poisson hyperplane process in R d . (a) The hyperplanes of b X are a.s. in translational general position. (b) If the spherical directional distribution ϕ of b X is subspace-free, then a.s. the hyperplanesof b X are in directionally general position.Proof. Let m ∈ N and A ∈ B ( H m ). We apply [14], Theorem 3.1.3 and Corollary 3.2.4, andthen (1), to obtain E X ( H ,...,H m ) ∈ b X m = A ( H , . . . , H m ) (3)= Z H m A ( H , . . . , H m ) b Θ m (d( H , . . . , H m ))= b γ m Z ( S d − ) m Z ∞−∞ · · · Z ∞−∞ A ( H ( u , τ ) , . . . , H ( u m , τ m )) d τ · · · d τ m ϕ m (d( u , . . . , u m )) . (a) Let k ∈ { , . . . , d − } be given. Define A ∈ B ( H d − k +1 ) by A := { ( H , . . . , H d − k +1 ) ∈ H d − k +1 : dim ( H ∩ · · · ∩ H d − k +1 ) = k } . Suppose that, for fixed u , . . . , u d − k +1 and suitable τ , . . . , τ d − k +1 ,dim ( H ( u , τ ) ∩ · · · ∩ H ( u d − k +1 , τ d − k +1 )) = k. Then we can assume, after renumbering, thatdim ( H ( u , τ ) ∩ · · · ∩ H ( u d − k , τ d − k )) = k, and we have H ( u , τ ) ∩ · · · ∩ H ( u d − k , τ d − k ) ∩ H ( u d − k +1 , τ ) = ∅ for all τ except one value. Therefore Z ∞−∞ A ( H ( u , τ ) , . . . , H ( u d − k , τ d − k ) , H ( u d − k +1 , τ )) d τ = 0 . Relation (3) (with m = d − k + 1) gives the assertion.(b) Suppose that ϕ is subspace-free. Let A be the set of all ( d + 1)-tuples ( H , . . . , H d +1 ) ∈H d +1 with linearly dependent normal vectors. Then A = S d +1 i =1 A i , where A i is the set of all( d + 1)-tuples ( H , . . . , H d +1 ) ∈ H d +1 for which the normal vector of H i is in the linear hullof the normal vectors of the remaining hyperplanes. For fixed u , . . . , u d , τ , . . . , τ d +1 we have Z S d − A d +1 ( H ( u , τ ) , . . . , H ( u d , τ d ) , H ( u d +1 , τ d +1 )) ϕ (d u d +1 ) = 0 , since the integrand is zero unless u d +1 lies in some fixed ( d − A , . . . , A d . Now relation (3) for m = d + 1, together with Fubini’s theorem,gives the assertion. 4he stationary Poisson hyperplane process b X induces a tessellation X of R d into convexpolytopes. In particular, for each k ∈ { , . . . , d } it induces the particle process X ( k ) of its k -faces (thus, X ( d ) = X is the process of its ‘cells’).Generally, let Y be a stationary particle process in a Borel subset K ′ of K . A centerfunction on K ′ is a measurable map c : K ′ → R d satisfying c ( K + t ) = c ( K ) + t for t ∈ R d and c ( K ) = o ⇒ c ( αK ) = o if α ≥
0, for all K ∈ K ′ . An often used center function is thecircumcenter (the center of the smallest ball containing the set), but later we shall need adifferent one, not defined on all of K . We define K ′ c := { K ∈ K ′ : c ( K ) = o } . If the intensitymeasure Θ = E Y satisfies Θ = 0 and Θ( { K ∈ K ′ : K ′ ∩ C = ∅} ) < ∞ for every compactsubset C ⊂ R d , then there are a unique number γ > Q on K ′ c such that Z K ′ f dΘ = γ Z K ′ c Z R d f ( K + x ) λ d (d x ) Q (d K ) (4)for every nonnegative, measurable function f on K ′ . We refer to [14, Section 4.1], but pointout that the definition of a center function has slightly been changed. The measure Q , calledthe grain distribution of Y , depends on the choice of the center function, although this is notshown in the notation. The intensity γ is independent of the choice of the center function,as follows, e.g., from [14], Theorem 4.1.3(b) with ϕ ≡
1. Campbell’s theorem now reads E X K ∈ Y f ( K ) = Z K ′ f dΘ . (5)Let A ∈ B ( K ′ ) and B ∈ B ( R d ) with 0 < λ d ( B ) < ∞ . Applying (4) and (5) with f ( K ) = A ( K − c ( K )) B ( c ( K )), we get γ Q ( A ) = 1 λ d ( B ) E X K ∈ X, c ( K ) ∈ B A ( K − c ( K )) . (6)For the process X ( k ) of k -faces of the tessellation X induced by b X , we denote the intensityby γ ( k ) and the intensity measure by Q ( k ) . This intensity measure satisfies the assumptionmade above (as follows, e.g., from [14, Thm. 10.3.4]), hence the previous results can beapplied. By P k ⊂ K we denote the set of k -dimensional polytopes; then P k is a Borel subsetof K . Since X ( k ) is a particle process in P k , for its description we need only center functionson P k . If such a center function is chosen, we denote by Z ( k ) the typical k -face of X , whichis defined as a random polytope with distribution Q ( k ) . Thus, for A ∈ B ( P k ) and B ∈ B ( R d )with λ d ( B ) = 1, we have γ ( k ) Q ( k ) ( A ) = E X K ∈ X ( k ) A ( K − c ( K )) B ( c ( K )) . (7) Later arguments require to base the investigation of the typical k -face on the incenter asa center function. The incenter of a k -dimensional convex body is the center of its largestinscribed k -dimensional ball, if that is unique. Since the latter is not always the case, someprecautions are necessary. The inradius of a k -dimensional convex body is the radius of alargest k -dimensional ball contained in the body.5 emma 2. Let P ∈ P be a d -polytope with the property that its facet hyperplanes are indirectionally general position. Then P has a unique inball.Proof. Let B be an inball of P . Let u , . . . , u k be the outer unit normal vectors of the facetsof P that touch B , and let C be the positive hull of these vectors. If the closed convex cone C does not contain a linear subspace of positive dimension, then there is a vector t in theinterior of the polar cone of C . This vector satisfies h t, u i i < i = 1 , . . . , k and hence B + εt ⊂ int P for some ε >
0, a contradiction. Thus, C contains a subspace L of dimension k >
0. Since L is positively spanned by normal vectors of P , it contains at least k + 1 suchvectors, and these are linearly dependent. This shows that k = d . But if the facets of P touching B have normal vectors which positively span R d , then these facets fix the ball B against translations, hence it is the unique inball of P .The notions of general position immediately carry over to ( k − k -dimensional affine subspace L , since after choosing an origin in L , we can view L as a k -dimensional real vector space.Let u = ( u , . . . , u d +1 ) be a ( d + 1)-tuple of unit vectors in general position, that is,any d of the vectors are linearly independent. For given k ∈ { , . . . , d − } , we define the k -dimensional subspace L u = (lin { u k +2 , . . . , u d +1 } ) ⊥ . We have orthogonal decompositions u j = v j + w j with v j ∈ L u , w j ∈ L ⊥ u , j = 1 , . . . , k + 1 . Here v j = o , since otherwise u j ∈ L ⊥ u = lin { u k +2 , . . . , u d +1 } , a contradiction. We claimthat any k vectors of v , . . . , v k +1 are linearly independent. Otherwise, some k of them, say v , . . . , v k , span a proper subspace E of L u . But then u , . . . , u k , u k +2 , . . . , u d +1 span only aproper subspace of R d , which is a contradiction.Now we can prove the following lemma. Lemma 3.
Let H , . . . , H d +1 be hyperplanes in general position. Let k ∈ { , . . . , d − } andlet L := T d +1 i = k +2 H i . Let h j := H j ∩ L for j = 1 , . . . , k + 1 . Then h , . . . , h k +1 are in generalposition with respect to L .Proof. Let u i be a unit normal vector of H i , for i = 1 , . . . , d + 1. Then u = ( u , . . . , u d +1 )is in general position, and we define L u and the vectors v , . . . , v k +1 as above. The affinesubspace L is a translate of L u . Let x, y ∈ h j . Then h x − y, u j i = 0 and h x − y, w j i = 0, hence h x − y, v j i = 0. It follows that v j is a normal vector of h j with respect to L . Thus, h , . . . , h k +1 are in directionally general position. They are also in translationally general position, sincethey do not have a common point, because H , . . . , H d +1 do not have a common point.The integral transform of the subsequent lemma is an extension of Theorem 7.3.2 in [14](and also a correction—a factor 2 d +1 was missing there). The extension consists in admittingmore general measures (not necessarily motion invariant) and using incenter and inradius ofalso lower-dimensional simplices. The lemma requires some preparations.Let H , . . . , H d +1 be hyperplanes in general position, and let k ∈ { , . . . , d − } . Let u i be a unit normal vector of H i , for i = 1 , . . . , d + 1. We define L u , L , h , . . . , h k +1 , and thevectors v , . . . , v k +1 as above. Thus, the k -flat L , the ( k − h , . . . , h k +1 and theirnormal vectors v , . . . , v k +1 in L u depend on H , . . . , H d +1 and on the number k .6ince h , . . . , h k +1 are in general position in L , they determine a unique k -simplex △ ( H , . . . , H d +1 ) ⊂ L such that h , . . . , h k +1 are the facet hyperplanes of △ (in L ). Wedenote by z ( H , . . . , H d +1 ) its incenter and by r ( H , . . . , H d +1 ) its inradius. The dependenceon k is not shown in the notation, since k is fixed.We denote by P k ⊂ ( S d − ) d +1 the set of all ( d +1)-tuples ( u , . . . , u d +1 ) in general positionsuch that the vectors v , . . . , v k +1 positively span the subspace L u .For u = ( u , . . . , u d +1 ) ∈ P k and for z ∈ R d and r ∈ R + , we define t j ( u , z, r ) := ( h z, u j i + r k u j | L u k if j ∈ { , . . . , k + 1 } , h z, u j i if j ∈ { k + 2 , . . . , d + 1 } , where u j | L u is the image of u j under orthogonal projection to L u . The mapping( z, r ) ( t ( u , z, r ) , . . . , t d +1 ( u , z, r ))has Jacobian D k ( u ) := det h u , e i · · · h u , e d i k u | L u k ... ... ... h u k +1 , e i · · · h u k +1 , e d i k u k +1 | L u kh u k +2 , e i · · · h u k +2 , e d i h u d +1 , e i · · · h u d +1 , e d i , where ( e , . . . , e d ) is an orthonormal basis of R d . Lemma 4.
Suppose that the spherical directional distribution ϕ of b X is subspace-free. Let k ∈ { , . . . , d } . If f : H d +1 → R is a nonnegative, measurable function, then Z H d +1 f d b Θ d +1 = 2 k +1 b γ d +1 Z P k Z R d Z ∞ f ( H ( u , t ( u , z, r )) , . . . , H ( u d +1 , t d +1 ( u , z, r ))) × d r λ d ( z ) D k ( u ) ϕ d +1 (d u ) . Proof.
In the following proof, we assume that k ≤ d −
1. The proof for the case k = d requiresonly the replacement of L u , L by R d , of v i by u i , and the obvious modifications. We have D d ( u ) = d !∆( u ), where ∆( u ) is the volume of the convex hull of u , . . . , u d +1 .For ( u i , τ i ) ∈ S d − × R , write f ′ (( u , τ ) , . . . , ( u d +1 , τ d +1 )) := f ( H ( u , τ ) , . . . , H ( u d +1 , τ d +1 )) . According to (1), we can write Z H d +1 f d b Θ d +1 = b γ d +1 Z ( S d − × R ) d +1 f ′ d( ϕ ⊗ λ ) d +1 , where λ denotes Lebesgue measure on R . Since ϕ is subspace-free, ( ϕ ⊗ λ ) d +1 -almostall ( d + 1)-tuples (( u , τ ) , . . . , ( u d +1 , τ d +1 )) ∈ ( S d − × R ) d +1 have the property that the7yperplanes H i = H ( u i , τ i ), i = 1 , . . . , d + 1, determine a unique k -simplex △ ( H , . . . , H d +1 )as above. The vectors v , . . . , v k +1 , defined as above, are the normal vectors of the facets of △ ,either inner or outer normal vectors. We define A as the set of all (( u , τ ) , . . . , ( u d +1 , τ d +1 )) ∈ ( S d − × R ) d +1 for which the hyperplanes H ( u , τ ) , . . . , H ( u d +1 , τ d +1 ) are in general positionand determine a k -simplex △ that has the vectors v , . . . , v k +1 as outer normal vectors. For E = ( ǫ , . . . , ǫ k +1 ) ∈ {− , } k +1 , let T E : ( S d − × R ) d +1 → ( S d − × R ) d +1 be defined by T E (( u , τ ) , . . . , ( u d +1 , τ d +1 )):= (( ǫ u , ǫ τ ) , . . . , ( ǫ k +1 u k +1 , ǫ k +1 τ k +1 ) , ( u k +2 , τ k +2 ) , . . . , ( u d +1 , τ d +1 )) . Then the sets T E ( A ), E ∈ {− , } k +1 , are pairwise disjoint and cover ( S d − × R ) d +1 up to aset of ( ϕ ⊗ λ )-measure zero. We have f ′ ◦ T E = f ′ . Since the pushforward of ( ϕ ⊗ λ ) d +1 under T E is the same measure (since ϕ and λ are even), we have Z ( S d − × R ) d +1 f ′ d( ϕ ⊗ λ ) d +1 = X E Z T E ( A ) f ′ d( ϕ ⊗ λ ) d +1 = 2 k +1 Z A f ′ d( ϕ ⊗ λ ) d +1 . This gives Z H d +1 f d b Θ d +1 = b γ d +1 Z ( S d − × R ) d +1 f ′ d( ϕ ⊗ λ ) d +1 = 2 k +1 b γ d +1 Z A f ′ d( ϕ ⊗ λ ) d +1 = 2 k +1 b γ d +1 Z P k Z R d +1 f ( H ( u , τ ) , . . . , H ( u d +1 , τ d +1 )) × A (( u , τ ) , . . . , ( u d +1 , τ d +1 )) d( τ , . . . , τ d +1 ) ϕ d +1 (d u ) . Let (( u , τ ) , . . . , ( u d +1 , τ d +1 )) ∈ A . The hyperplanes H ( u , τ ) , . . . , H ( u d +1 , τ d +1 ) deter-mine the simplex △ , with incenter z and inradius r . Let x j ∈ L u be the point where H ( u j , τ j )touches the inball of △ . Then h x j − z, v j / k v j ki = r and τ j = h x j , u j i = h z, u j i + h x j − z, u j i = h z, u j i + h x j − z, v j + w j i = h z, u j i + h x j − z, v j i = h z, u j i + r k v j k = t j ( u , z, r ) . Thus, we see that the mapping( z, r, u ) (( u , t ( u , z, r )) , . . . , ( u d +1 , t d +1 ( u , z, r )))maps R d × R > × P k bijectively onto A . As mentioned, for fixed u ∈ P k , the mapping( z, r ) ( t ( u , z, r ) , . . . , t d +1 ( u , z, r ))has Jacobian D k ( u ). Therefore, the transformation formula for multiple integrals, appliedfor each fixed u ∈ P k to the inner integral, gives the assertion.8 The distribution of the typical k -face Our next aim is to extend [14, Thm. 10.4.6], giving a representation for the distribution ofthe typical k -face Z ( k ) with respect to the incenter as a center function.For u ∈ P k , we define T k ( u ) := L u ∩ k +1 \ i =1 H − ( u i , k u i | L u k ) . This is a k -dimensional simplex in the subspace L u , with incenter o and inradius 1.For a k -dimensional affine subspace L , a point z ∈ L and a number r >
0, we define B o ( L, z, r ) as the relative interior (with respect to L ) of the k -dimensional ball in L withcenter z and radius r .Of the Poisson hyperplane process b X we assume that its spherical directional distributionis subspace-free. Then the hyperplanes of b X are a.s. in general position. For the typical k -face Z ( k ) of the tessellation X , we take the incenter as the center function c . This is possibleby Lemmas 2 and 3. Let F ∈ X ( k ) be a k -face of the tessellation X , and let L be its affinehull. Then there are d − k hyperplanes of b X whose intersection is L . They can be numberedin ( d − k )! ways. The face F has a k -dimensional inball. It determines k + 1 hyperplanes of b X whose intersections with L touch the inball. These hyperplanes can be numbered in ( k + 1)!ways. Therefore, (7) gives, for A ∈ B ( K ), the subsequent relation. In the following formulas, L, △ , z, r are functions of the hyperplanes H , . . . , H d +1 in general position (and the givennumber k ∈ { , . . . , d − } ), namely L = d +1 \ i = k +2 H i , △ is the simplex in L determined by L ∩ H , . . . , L ∩ L k +1 , the point z is its incenter, andthe number r is its inradius. The dependence of the functions L, △ , z, r on the hyperplanes H , . . . , H d +1 is not shown in the following formulas, but has to be kept in mind. We recallthat H − x denotes the closed halfspace bounded by the hyperplane H that contains x , providedthat x / ∈ H .We get by (7), using the unit cube C d as the set B , γ ( k ) Q ( k ) ( A ) = E X K ∈ X ( k ) ( K − c ( K )) C d ( c ( K ))= 1( k + 1)!( d − k )! E X ( H ,...,H d +1 ) ∈ b X d +1 = n b X ∩ [ B o ( L, z, r )] H = ∅ o C d ( z ) × \ H ∈ b X \ [ B o ( L,z,r )] H H − z − z ∈ A . Application of the Mecke formula (Corollary 3.2.3 of [14]) yields γ ( k ) Q ( k ) ( A ) = 1( k + 1)!( d − k )! Z H d +1 E n b X ∩ [ B o ( L, z, r )] H = ∅ o C d ( z ) × \ H ∈ b X \ [ B o ( L,z,r )] H (cid:0) △ ∩ H − z (cid:1) − z ∈ A b Θ(d( H , . . . , H d +1 )) . b X ∩ [ B o ( L, z, r )] H and b X \ [ B o ( L, z, r )] H are independent. Further, denotingby L o the subspace which is a translate of L and by B ( L o ) the k -dimensional closed ball in L o with center o and radius 1, we have P (cid:16) b X ∩ [ B o ( L, z, r )] H = ∅ (cid:17) = P (cid:16) b X ∩ [ rB ( L o )] H = ∅ (cid:17) = e − b γr Φ( B ( L o )) . Thus, we obtain γ ( k ) Q ( k ) ( A ) = 1( k + 1)!( d − k )! Z H d +1 e − b γr Φ( B ( L o )) C d ( z ) × P \ H ∈ b X \ [ B o ( L,z,r )] H (cid:0) △ ∩ H − z (cid:1) − z ∈ A b Θ d +1 (d( H , . . . , H d +1 ))= 1( k + 1)!( d − k )! Z H d +1 e − b γr Φ( B ( L o )) C d ( z ) × P \ H ∈ b X \ [ B o ( L o ,o,r )] H (cid:0) ( △ − z ) ∩ H − o (cid:1) ∈ A b Θ d +1 (d( H , . . . , H d +1 )) , where in the last step the stationarity of b X was used.Now we apply Lemma 4. We recall that L, △ , z, r in the integrand above are functions of H , . . . , H d +1 . In particular, we have L o ( H ( u , τ ) , . . . , H ( u d +1 , τ d +1 )) = L u , u = ( u , . . . , u d +1 ) . If we first denote (for fixed u = ( u , . . . , u d +1 )) the integration variables of the inner integralsin Lemma 4 by z ′ and r ′ , we have z ( H ( u , t ( u , z ′ , r ′ ) , . . . ) = z ′ ,r ( H ( u , t ( u , z ′ , r ′ ) , . . . ) = r ′ , △ ( H ( u , t ( u , z ′ , r ′ ) , . . . ) − z ( H ( u , t ( u , z ′ , r ′ ) , . . . ) = r ′ T k ( u ) . Therefore, we obtain the following theorem.
Theorem 1.
Suppose that the spherical directional distribution ϕ of b X is subspace-free. Let k ∈ { , . . . , d } . With respect to the incenter as center function, the distribution Q ( k ) of thetypical k -face Z ( k ) of the tessellation X induced by b X is given by Q ( k ) ( A )= 2 k +1 ( k + 1)!( d − k )! b γ d +1 γ ( k ) Z P k Z ∞ e − b γr Φ( B ( L u )) P \ H ∈ b X \ [ B o ( L u ,o,r )] H H − o ∩ rT k ( u ) ∈ A × d rD k ( u ) ϕ d +1 (d u ) for A ∈ B ( K ) . The shape of small k -faces As we want to find the limit distribution of the shape of small k -faces of the tessellation X , weneed a measurement for size and a notion of shape. We consider P k , the set of k -dimensionalpolytopes in R d , for a given number k ∈ { , . . . , d } , and the subset P ′ k of polytopes witha unique ( k -dimensional) inball. A size functional for P k is a real function on K that isincreasing under set inclusion, homogeneous of degree one, continuous (with respect to theHausdorff metric), and positive on k -dimensional sets. The homogeneity of degree one isnot a restriction against the formerly required homogeneity of some degree m >
0, since ahomogeneous functional ψ of degree m may be replaced by ψ /m . Examples of size functionalsfor P k are V /kk , where V k is the k -dimensional volume, the mean width, the circumradius,or the diameter. On P ′ k we use the incenter as center function, denoted by c .The shape of P ∈ P ′ k , is obtained by normalization, s c ( P ) := 1Φ( P ) ( P − c ( P )) , where Φ is the hitting functional defined in Section 2. The shape space P k,c , defined by P k,c = s c ( P ′ k ) = { P ∈ P ′ k : c ( P ) = o, Φ( P ) = 1 } , is a Borel subset of P . By f r ( P ) we denote the number of r -dimensional faces of a polytope P . Theorem 2.
Suppose that the spherical directional distribution of b X is subspace-free. Let k ∈ { , . . . , d } . Let Σ be a translation invariant size functional for P k . Then the typical cell Z ( k ) of the tessellation X satisfies lim a → P (cid:16) s c ( Z ( k ) ) ∈ S | Σ( Z ( k ) ) < a (cid:17) = ξ k,ϕ, Σ ( S ) ξ k,ϕ, Σ ( P k,c ) for S ∈ B ( P k,c ) , where the measure ξ k,ϕ, Σ on B ( P k,c ) is defined by ξ k,ϕ, Σ := Z P k { s c ( T k ( u )) ∈ ·} D k ( u )Σ( T k ( u )) ϕ d +1 (d u ) . In particular, lim a → P (cid:16) f k − ( Z ( k ) ) = k + 1 | Σ( Z ( k ) ) < a (cid:17) = 1 . Proof.
Let S ∈ B ( P k,c ) and a > A = { P ∈ P ′ k : f k − ( P ) = k + 1 , s c ( P ) ∈ S, Σ( P ) < a } . We have f k − \ H ∈ b X \ [ B o ( L u ,o,r )] H H − o ∩ rT k ( u ) = k + 1 ⇔ b X ([ rT k ( u )] H \ [ B o ( L u , o, r )] H ) = 011nd P (cid:16) b X ([ rT k ( u )] H \ [ B o ( L u , o, r )] H ) = 0 (cid:17) = P (cid:16) b X ([ rT k ( u )] H \ [ rB ( L u )] H ) = 0 (cid:17) = e − b γr [Φ( T k ( u )) − Φ( B ( L u ))] . Hence, from Theorem 1 we get P (cid:16) f k − ( Z ( k ) ) = k + 1 , s c ( Z ( k ) ) ∈ S, Σ( Z ( k ) ) < a (cid:17) = 2 k +1 ( k + 1)!( d − k )! b γ d +1 γ ( k ) Z P k Z ∞ e − b γr Φ( T k ( u )) { Σ( rT k ( u )) < a } d r × { s c ( T k ( u )) ∈ S } D k ( u ) ϕ d +1 (d u ) . By the mean value theorem, the inner integral is equal to a Σ( T k ( u )) e − b γr ( u )Φ( T k ( u )) with an intermediate value r ( u ) satisfying0 ≤ r ( u ) ≤ a Σ( T k ( u )) ≤ a Σ( B ( L u )) ≤ Ca, wnere C is a constant independent of u . Here we have used that B ( L u ) ⊂ T k ( u ) and that Σis increasing, finally that Σ is continuous, hence u Σ( B ( L u )) attains a minimum, which ispositive by the assumptions on the size functional. It follows thatlim a → a − P (cid:16) f k − ( Z ( k ) ) = k + 1 , s c ( Z ( k ) ) ∈ S, Σ( Z ( k ) ) < a (cid:17) = 2 k +1 ( k + 1)!( d − k )! b γ d +1 γ ( k ) Z P k { s c ( T k ( u )) ∈ S } D k ( u )Σ( T k ( u )) ϕ d +1 (d u ) . (8)Let r k ( K ) denote the inradius of a k -dimensional convex body K ⊂ L u . Then r k ( K ) B ( L u ) + t ⊆ K for a suitable vector t , hence r k ( K )Σ( B ( L u )) ≤ Σ( K ). Thus,Σ( Z ( k ) ) < a ⇒ r k ( Z ( k ) ) < a Σ( B ( L u )) . Therefore, P (cid:16) f k − ( Z ( k ) ) > k + 1 , Σ( Z ( k ) ) < a (cid:17) ≤ P (cid:18) f k − ( Z ( k ) ) > k + 1 , r k ( Z ( k ) ) < a Σ( B ( L u )) (cid:19) . With this, Theorem 1 gives P (cid:16) f k − ( Z ( k ) ) > k + 1 , Σ( Z ( k ) ) < a (cid:17) ≤ k +1 ( k + 1)!( d − k )! b γ d +1 γ ( k ) Z P k Z ∞ e − b γr Φ( B ( L u )) × P f k − \ H ∈ b X \ [ B o ( L u ,o,r )] H H − o ∩ rT k ( u ) > k + 1 × (cid:26) r < a Σ( B ( L u )) (cid:27) d rD k ( u ) ϕ d +1 (d u ) . u ∈ P k , the probability in the integrand tends to zero as r →
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