A Schlafli-type formula for polytopes withcurved faces and its application to the

Kneser-Poulsen conjecture

Balazs Csikos ∗

Eotvos UniversityBudapest

Abstract

We prove a Schlafli-type formula for polytopes with curved faceslying in pseudo-Riemannian Einstein manifolds. This formula is ap-plied then to the Kneser-Poulsen conjecture claiming that the volumeof the union of some balls can not increase when the balls are rear-ranged in such a way that the distances between the centers decrease.

1 Introduction

Let P1, . . . , Pk and Q1, . . . , Qk be two k-tuples of points in the Euclidean spaceRn. T. Poulsen [16], M. Kneser [14] and H. Hadwiger [13] asked wether theinequalities d(Pi, Pj) ≥ d(Qi, Qj) for all 1 ≤ i < j ≤ k imply that

Vn

( k⋃i=1

B(Pi, r))≥ Vn

( k⋃i=1

B(Qi, r)),

where Vn is the n-dimensional volume, r is an arbitrary positive number,B(P, r) is the ball of radius r around P .

Though this question is still open for n ≥ 3, there have been many papersdevoted to the verification of the conjecture and its generalizations in special

∗Supported by the Hungarian National Science and Research Foundation OTKAT047102.Keywords: Schlafli formula, pseudo-Riemannian Einstein manifold, Kneser-Poulsen con-jecture, CSG.MSC2000 Subject Classification: 53C21, 53C50, (51M25, 51M16).

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cases. The first series of papers [5], [7], [2], [8], [12], [11], [9] considered thespecial case, when the balls are moved smoothly in such a way that during themotion all the center-center distances change in a monotonous way. In thisspecial case monotonicity results were obtained not only for the volume of theunion of congruent balls of the Euclidean space, but also for the volume offlowers, i.e. domains in the hyperbolic, Euclidean, or spherical space obtainedfrom not necessarily congruent balls with the help of the operations ∩ and∪ (see [9]).

It is not true in general that if P1, . . . , Pk and Q1, . . . , Qk are two configu-rations in Rn, and d(Qi, Qj) ≤ d(Pi, Pj) for all 1 ≤ i < j ≤ k, then there arecontinuous curves γi : [0, 1] → Rn connecting γi(0) = Pi to γi(1) = Qi suchthat the distances d(γi(t), γj(t)) weakly decrease, however, one can alwaysconstruct such curves in R2n. This simple observation, called the LeapfrogLemma yields that if we want to show that a certain function on k-tuples ofpoints in Rn does not increase when the points are contracted, it is enoughto find an extention of that function to arbitrary k-tuples in R2n which doesnot increase when the points are contracted continuously. This idea has beenfruitfully applied in [1], [6], [3], [4]. For instance in the remarkable paper [3],K. Bezdek and R. Connelly could prove the Kneser-Poulsen conjecture in theplane using this method.

When the balls are moved smoothly, the change of the volume of theunion of the balls can be controlled by a formula expressing the variationof the volume as a linear combination of the derivatives of the center-centerdistances with coefficients equal to the volumes of the walls of the Dirichlet-Voronoi decomposition of the union of the balls (see [8]). This formula andits extension for flowers obtained in [9] resemble the classical Schlafli formulaexpressing a multiple of the variation of the volume of a simplex in the hy-perbolic or spherical space as a linear combination of the derivatives of thedihedral angles with coefficients equal to the volumes of the 2-codimensionalfaces. The author’s primary motivation for the study of Schlafli type for-mulae was to find a common root of these formulae. Roughly speaking,simplices and unions of balls belong to the class of polytopes with curvedfaces. The family of polytopes with curved faces contains also compact do-mains bounded by smooth hypersurfaces. For such domains lying in Einsteinmanifolds, I. Rivin and J.-M. Schlenker proved a Schlafli type formula in [17].Analysing the sketch of the proof of the formula in [17] the author found aSchlafli-type formula for polytopes with curved faces in Riemannian Einsteinmanifolds which contained both the classical Schlafli formula and the for-mula of Rivin and Schlenker as a special case. This formula and some ofits applications to the Kneser-Poulsen conjeture was presented at the BolyaiBicentennial Conference in Budapest in 2002. Right after the conference the

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author received a preprint from R. Souam devoted to the proof of a Schlaflitype formula for piecewise smooth immersions of simplicial complexes intopseudo-Riemannian Einstein manifolds. The Schlafli-type formula we presentin this paper is an extension of the formula presented at the Bolyai Confer-ence to the pseudo-Riemannian case. Our formula is proved in a differentsetting and is slightly more general than that of R. Souam, which appearedrecently in [18]. The main difference is that in Souam’s formula the infinites-imal variation of the polytope is assumed to be continuous on the boundary,while in ours the faces of the polytope are deformed separately and the in-finitesimal variations of the faces may not coincide on the walls between thefaces. The formula in this paper contains exactly the same terms as Souam’sformula together with an extra term which is related to the discontinuity ofthe infinitesimal variations of the faces along the walls between the faces.This extension of Souam’s formula turns out to be quite useful when the for-mula is applied to the union of some smoothly moving balls. Since the ballsmove rigidly, we can choose Killing fields for the infinitesimal variations ofthe faces in our formula which simplifies the evaluation of the formula. How-ever, in general it is impossible to choose a continuous infinitesimal variationcompatible with the deformation of the boundary the restrictions of whichonto the faces of the union are Killing fields.

The paper has the following structure. In Section 2 we give an exact def-inition of a polytope with curved faces in a manifold, and we introduce somebasic notions related to polytopes (faces, walls between the faces, dihedralangles, variation of a polytope and compatible infinitesimal variations, etc.).Section 3 contains some variational formulae which culminate in the Schlafli-type formula in Theorem 2. In Section 3, our formula is specialized for poly-topes made of some rigidly moving balls. The relation of the resulting formulato the Kneser-Poulsen conjecture is illuminated by an “Archimedean” for-mula for solids of revolution (Theorem 5). The main result of this section isTheorem 6 extending some of the results of [8], [9] and [3].

2 Polytopes with curved faces

Intuitively, a polytope with curved faces is a compact domain in a mani-fold, bounded by a finite number of smooth hypersurfaces. There are severalmathematical models grasping this intuitive concept. We shall use the ap-proach of Constructive Solid Geometry (CSG) and define polytopes withcurved faces as solids that can be obtained by applying regularized Booleanset operations to regular domains of a manifold.

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2.1 CSG representations of polytopes with curved faces

Let X be a topological space and denote by CX the distributive lattice ofclosed subsets of X, by IX the ideal in CX formed by the nowhere denseclosed subsets of X. The factor lattice CX/IX is a Boolean algebra since aclosed set intersects the closure of its complement in a nowhere dense set,thus we have complements in CX/IX .

A regular closed subset of X is a subset, which is equal to the closureof its interior. There is a regularization operator ρX : PX → RX from thepower set PX of X to the set RX of regular closed subsets of X defined byρX(A) = int(A). Every (mod IX) congruence class in CX contains a uniqueregular closed set, which can be obtained by the regularization of any of theclosed sets belonging to the equivalence class, thus, there is a natural bijec-tion between the Boolean algebra CX/IX and the set RX . Transferring theBoolean algebra structure from CX/IX to RX we obtain a Boolean algebrastructure on the set of regular closed subsets of X. The induced operationson RX are the regularized Boolean operations

A ∪∗ B = ρX(A ∪B), A ∩∗ B = ρX(A ∩B), A \∗ B = ρX(A \B).

A Boolean expression f is a formal expression (a finite sequence of sym-bols) such that f is either a symbol for a single variable or an expression ofthe form (f1 ∗ f2), where f1 and f2 are Boolean expressions, ∗ is one of thesymbols ∪, ∩, \. Each Boolean expression f corresponds to a regularizedBoolean expression f ∗ obtained from f by replacing each operation sym-bol by the symbol of the corresponding regularized operation. Two Booleanexpressions are considered to be the same if and only if they are identi-cal as sequences of symbols. Thus, x, (x ∪ x), ((x ∪ y) ∩ x) are differentBoolean expressions. If f(x1, . . . , xk) is a Boolean expression in k variables,A1, . . . , Ak ∈ PX are subsets of X, then we denote by f(A1, . . . , Ak) andf ∗(A1, . . . , Ak) the evaluation of the expressions on the sets A1, . . . , Ak. Ifthe sets A1, . . . , Ak are contained in a subset Y of X, then we denote byf ∗Y (A1, . . . , Ak) the evaluation of f ∗ in Y computed with operations regu-larized by ρY instead of ρX . The following proposition is a straightforwardcorollary of the definitions.

Proposition 1. If A1, . . . , Ak ∈ RX , B1, . . . , Bk ∈ PX are subsets suchthat the symmetric differences Ai4Bi are nowhere dense, f(x1, . . . , xk) is anarbitrary Boolean expression, f ∗(x1, . . . , xk) its regularization, then we have

f ∗(A1, . . . , Ak) = ρX(f(A1, . . . , Ak)) = ρX(f(B1, . . . , Bk)).

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Let M be an n-dimensional smooth manifold. A regular domain in M isa subset of the form {p ∈ M | f(p) ≤ 0}, where f : M → R is a smooth func-tion on M , such that 0 is a regular value of f . (Another equivalent definitioncan be found in 4.8. [19].) M and ∅ are regular domains in M . Regulardomains are n-dimensional manifolds with or without boundary embeddedas regular closed sets in M . The boundary of a regular domain is eitherempty, or it is a smooth hypersurface in M .

A CSG solid in a manifold M is a compact set which can be obtained asthe evaluation of a regularized Boolean expression on some regular domains ofM . A CSG representation of a CSG solid P is a regularized Boolean expres-sion f ∗(x1, . . . , xk) together with a collection of regular domains P1, . . . , Pk

such that P = f ∗(P1, . . . , Pk).In Constructive Solid Geometry, the regular domains P1, . . . , Pk are called

the primitives from which P is built, and CSG representations are depictedby a rooted binary tree, called CSG tree, in which the leaves are markedwith primitives, and the internal nodes are marked with regularized Booleanoperations (see Fig. 1). The CSG representation of a CSG solid is not unique.

Figure 1: Example of a CSG tree

Definition 2.1. Let s ≥ 1 be a natural number. A CSG solid P with a

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fixed CSG representation f ∗(P1, . . . , Pk) will be called s-transversal if thefollowing two properties hold:

(i) each of the variables x1, . . . , xk occurs in f ∗(x1, . . . , xk) exactly once;

(ii) any l ≤ s of the hypersurfaces ∂Pi, (1 ≤ i ≤ k) intersect transver-sally. (We say that the hypersurfaces Σ1, . . . , Σl intersect transversallyif dim

⋂li=1 TpΣi = (n− l) for all p ∈

⋂li=1 Σi.)

Throughout this paper the term polytope with curved faces or simplypolytope will be used for 3-transversal CSG solids.

2.2 Faces of a polytope and the walls between them

Definition 2.2. Let P = f ∗(P1, . . . , Pk) be a 2-transversal CSG solid inM . The ith face Fi of P is the regularized contribution of the hypersurfaceΣi = ∂Pi to the boundary of P , that is Fi = ρΣi

(Σi ∩ ∂P ).

Remark that the definition above is slightly different from the ones com-monly used in CSG, where the faces are the (n − 1)-dimensional cells of aCW structure on the boundary.

Proposition 2. The faces of an s-transversal CSG solid with s ≥ 2 have thefollowing properties.

(i) Fi is an (n− 1)-dimensional (s− 1)-transversal CSG solid in Σi, thatcan be built up from the primitives Pj ∩ Σi, (1 ≤ j ≤ k), that is, thereis a regularized Boolean expression (∂if

∗)(x1, . . . , xk) in which each of itsvariables occurs exactly once and

Fi = ∂if∗Σi

(P1 ∩ Σi, . . . , Pk ∩ Σi).

∂if∗ depends only on f ∗ and can be defined recursively by the following rules:

∂ixi = xi and if g∗ and h∗ are Boolean expressions with single use of disjointsets of variables and xi is a variable of g∗, then

∂i(g∗ ∪∗ h∗) = ∂i(h

∗ ∪∗ g∗) = ∂i(g∗ \∗ h∗) = (∂ig

∗) \∗ h∗,

∂i(g ∩∗ h) = (∂ig) ∩∗ h, ∂i(h ∩∗ g) = ∂i(h \∗ g) = h ∩∗ (∂ig).

(ii) The union of all the faces equals the boundary of P .

Proof. (i) The proof of the first part of the proposition goes by induction onthe length of f ∗. Though it is a bit lengthy due to the several cases for thestructure of f ∗, it is quite straightforward, so we omit the details.

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(ii) Set Ni =⋃

1≤j≤kj 6=i

Σi ∩ Σj Denote by F ◦i the set of those boundary

points of P which belong only to Σi, that is,

F ◦i = (Σi ∩ ∂P ) \Ni.

F ◦i is open in Σi and therefore F ◦

i ⊂ Fi. We show that F ◦i is dense in

Fi. Indeed, if U ⊂ Σi is an arbitrary open neighborhood of a point p ∈ Fi,then since Fi is regular closed in Σi, U contains a non-empty open (in Σi)subset V ⊂ Fi ∩ U . Since V is (n − 1)-dimensional, while the intersectionsΣi ∩ Σj are (n − 2)-dimensional, V must have a point, which is not in Ni.Thus Fi = F ◦

i .Finally we claim that

⋃1≤i≤k F ◦

i is a dense subset of ∂P . To prove this,

take a point p ∈ ∂P and an arbitrary connected neighborhood U of it. SinceP is regular closed, the interior of P is dense in P , therefore U contains aninner point q of P . U must contain also a point r in the exterior of P , sincep is on the boundary. Connect q to r by a curve γ in U keeping away fromthe intersections Σi ∩Σj. This is possible since the intersections are (n− 2)-dimensional. As γ connects an interior point of P to a point in its exterior,γ must cross the boundary of P at a point s ∈ ∂P ∩ U . Since ∂P is coveredby⋃

i Σi, s is contained in Σi for an i. i is uniquely determined as γ missesthe intersections Σi ∩ Σj. Consequently, s ∈ U belongs to F ◦

i . We conclude

that ∂P =⋃k

i=1 F ◦i =

⋃ki=1 F ◦

i =⋃k

i=1 Fi.

Using the above notation and the terminology of Chp. XVII. of [15], theinterior of a 2-transversal CSG solid P is a manifold with singular boundary inM . As

⋃ki=1 F ◦

i consists of regular frontier points, the set of singular frontierpoints is covered by the union of the (n−2)-dimensional intersections Σi∩Σj.This means that Theorem 3.3 in Chp. XVII. [15] is applicable to P and yieldsthe following version of Stokes’ theorem:

Proposition 3. If M is oriented and the regular frontier of the 2-transversalCSG solid P is equipped with the usual induced orientation, then∫

P

dω =k∑

i=1

∫Fi

ω. (1)

for any differential (n− 1)-form ω on M .

If P is a polytope, i.e. 3-transversal, then we can apply Proposition 2 tothe faces Fi. The ith face of Fi is empty since ∂Σi

(Σi ∩ Pi) = ∅, the jth facefor j 6= i is the regularized contribution of Σi ∩ Σj to the boundary of theface Fi, i.e. the set Wij = ρΣij(Σi∩Σj)∩∂Σi

Fi. Wij is an (n−2)-dimensional

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CSG solid in Σi ∩ Σj. Comparision of the CSG representations of Wij andWji derived in the proof of Proposition 2 reveals that Wij = Wji. The solidWij will be called the wall between the faces Fi and Fj.

2.3 Polytopes in pseudo-Riemannian manifolds

Assume that the manifold M is equipped with a pseudo-Riemannian metric{, }. For a tangent vector X of M , we define the norm |X| and the sign ε(X)of X by |X| = |{X, X}|1/2 and ε(X) = sgn({X, X}) respectively.

Definition 2.3. A regular domain P of M will be called pseudo-Riemannianif the restriction of {, } onto the boundary Σ = ∂P is non-degenerate andhas constant index, i.e., if Σ is a pseudo-Riemannian submanifold of M . Apseudo-Riemannian s-transversal CSG solid in a pseudo-Riemannian man-ifold M is an s-transversal CSG solid P = f ∗(P1, . . . , Pk) for which theintersection of any l ≤ s of the hypersurfaces ∂Pi is a pseudo-Riemanniansubmanifold of M . In particular, a pseudo-Riemannian polytope is a pseudo-Riemannian 3-transversal CSG solid.

A pseudo-Riemannian regular domain P has a unique outer unit normalvector field NP ∈ Γ(TM |Σ) along Σ = ∂P . If P is defined by the inequalityf ≤ 0, where f is a smooth function on M and 0 is a regular value of f , thenNP is given by the equation

NP = ε(grad f)grad f

| grad f |.

A pseudo-Riemannian 2-transversal CSG solid P = f ∗(P1, . . . , Pk) has awell-defined smooth outer unit normal vector field along each face. If NPi

isthe outer unit normal vector field of Pi, then the outer unit normal vectorfield Ni of P along the face Fi is equal to (−1)siNPi

, where si is the numberof those subtractions in f where an expression containing the ith variable issubtracted from another expression.

Each face Fi of a pseudo-Riemannian polytope P = f ∗(P1, . . . , Pk), is apseudo-Riemannian 2-transversal CSG solid within the pseudo-Riemannianmanifold Σi and the faces of Fi are the walls between Fi and the other faces.Applying the above reasoning to Fi we obtain that Fi has a well definedsmooth outer unit normal vector field nij ∈ Γ(TΣi|Wij

) along the wall Wij.We are going to define the inner dihedral angle αij of the pseudo-Riemann-

ian polytope P along the wall Wij. αij is a smooth function on Wij computedas follows. Set εij = ε(nij) and εi = ε(Ni). For any p ∈ Wij, the vectorsnij(p) and Ni(p) form an orthonormal basis of the 2-dimensional orthogonal

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complement of TpWij in TpM . Since nji(p) is also in this orthogonal plane,we can write the vector field nji as a linear combination of nij and Ni:

nji = λεijnij + µNi, (2)

where λ and µ are smooth functions on the wall Wij. As |nji| = |nij| =|Ni| = 1 and {nij,Ni} = 0, equation (2) implies

εji = λ2εij + µ2εi. (3)

We have three possibilities for equation (3) depending on the signs εji, εij

and εi.If λ2 + µ2 = 1, then we define αij by the relations 0 < αij < 2π, εijλ =

cos ◦αij and εijµ = sin ◦αij.If λ2 − µ2 = 1, then we define αij by the equations |λ| = cosh ◦αij,

sgn(λ)µ = sinh ◦αij.Finally, if µ2 − λ2 = 1, then αij is the function determined by sgn(µ)λ =

sinh ◦αij, |µ| = cosh ◦αij.The ordered basis (nji(p),Nj(p)) is orthonormal and has the same orien-

tation as the ordered basis (Ni(p),nij(p)) for all p ∈ Wij. These conditionsallow us to compute that Nj is expressed as a linear combination of nji andNj in the following way

Nj = εiεjiµnij − εjiλNi. (4)

As a corollary of (2), (3) and (4) we obtain that

nij = λεjinji + µNj, (5)

which shows that αij = αji.

2.4 Variations of CSG solids

A CSG structure has a discrete component, the Boolean expression (or theCSG tree encoding it), which can not be deformed continuously. However,one can deform a CSG solid by deforming its primitives.

Throughout this section, I will denote an open interval around 0. WhenH : X × I → Y is an arbitrary homotopy and t ∈ I, Ht : X → Y will denotethe map defined by Ht(p) = H(p, t).

Definition 2.4. A variation of a regular domain P = P0 in M is a one-parameter family of regular domains Pt, t ∈ I, for which

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(i) there is a proper isotopy Φ : ∂P × I → M such that Φ0 is the inclusionof ∂P , and Φt(∂P ) = ∂Pt for all t ∈ I;

(ii) for any point p ∈ M the sets {t ∈ I | p ∈ int Pt} and {t ∈ I | p ∈ ext Pt}are open in I.

Less formally, during a variation of a regular domain the boundary istransformed by a proper isotopy and the domain is not allowed to flip fromone side of the boundary to the other.

The trace of a variation is the set P = {(x, t) ∈ M × I | x ∈ Pt}. Thetrace of a variation of a regular domain in M is a regular domain in M × I.

The variation of the boundary of P determines the variation of P uniquely,so studying variations of regular domains we may focus on the variation ofthe boundary.

A vector field X ∈ Γ(TM |Σ0) along Σ0 is said to be an infinitesimalvariation of Σ0, compatible with the given variation if one of the followingequivalent conditions is fulfilled:

• The isotopy Φ in Definition 2.4 (i) can be chosen in such a way that

Xp = ∂Φ(p,t)∂t

∣∣∣t=0

.

• The vector field (X, ∂/∂t) ∈ Γ(T (M × I)|Σ0×{0}) is tangential to theboundary of P.

The difference of two infinitesimal variations compatible with the same vari-ation can be an arbitrary tangential vector field along Σ0.

In particular, if P0 is a pseudo-Riemannian regular domain in a pseudo-Riemannian manifold M , NP is the outer unit normal vector field of P , thenevery infinitesimal variation X compatible with the given variation has thesame normal component νNP , where ν = ε(NP ){X,NP}.

Definition 2.5. A variation of a (pseudo-Riemannian) s-transversal CSGsolid

P = f ∗(P1, . . . , Pk) (6)

consists of variations P1,t, . . . , Pk,t, t ∈ I of the primitives P1, . . . , Pk respec-tively, such that

(i) Pt = f ∗(P1,t, . . . , Pk,t) is a (pseudo-Riemannian) s-transversal CSG solidfor all t ∈ I;

(ii) for any l ≤ s and 1 ≤ i1 < · · · < il ≤ k, there is a proper isotopyΦt, t ∈ I of

⋂lj=1 ∂Pij in M such that Φ0 is the inclusion map and

Φt(⋂l

j=1 ∂Pij) =⋂l

j=1 ∂Pij ,t for all t;

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(iii) there is a compact set K ⊂ M such that Pt ⊂ K for all t ∈ I.

Proposition 4. Let φ : M × I → R be an arbitrary continuous function,Pt = f ∗(P1,t, . . . , Pk,t) a variation of a CSG-solid in M . Then the integral∫

Pt

φ(p, t)dp (7)

taken with respect to the measure induced by the pseudo-Riemannian metricof M is a continuous function of t.

Proof. Denote by χt the characteristic function of Pt. For any number t0 ∈ I,the union of the boundaries of the primitives P1,t0 , . . . , Pk,t0 has measure 0.

However, for any point p ∈ M \⋃k

i=1 Pi,t0 , the function χt(p) is constant in asmall neighborhood of t0 and therefore limt→t0 φ(p, t)χt(p) = φ(p, t0)χt0(p) foralmost all p ∈ M . Since the characteristic functions χt (t ∈ I) have a commoncompact support, the statement follows from the Lebesgue lemma.

Let Pi be the trace of the variation of the primitive Pi. The trace ofthe variation of the solid is defined to be the set P = f ∗(P1, . . . , Pk). Thetrace of the variation of an s-transversal solid is not an s-transversal solid inM because of the lack of compactness. However, for any [a, b] ⊂ I, settingHa = {(p, t) ∈ M × I | t ≥ a} and Hb = {(p, t) ∈ M × I | t ≤ b}, we have:

Proposition 5. The truncated trace f ∗(P1, . . . , Pk) ∩∗ Ha ∩∗ Hb of the vari-ation of an s-transversal solid is an s-transversal solid in M × I.

Consider now a variation of a pseudo-Riemannian polytope (6) in a pseudo-Riemannian manifold (M, {, }). Denote by Σi the boundary of the primitivePi and let Xi ∈ Γ(TM |Σi

) be an infinitesimal variation of Σi compatiblewith the variation of Pi. A vector field X ∈ Γ(TM |Wij

) along the wall Wij

is said to be an infinitesimal variation of the wall Wij compatible with thevariation of the polytope if the isotopy Φt of the intersection Σi ∩Σj in part

(ii) of Definition 2.5 can be chosen in such a way that Xp = dΦt(p)dt

∣∣∣t=0

for all

p ∈ Wij.

3 Variational formulae

3.1 Variation of the volume

Consider a variation Pt = f ∗(P1,t, . . . , Pk,t) of a pseudo-Riemannian polytopein a pseudo-Riemannian manifold (M, {, }) and denote by V (t) the volumeof Pt with respect to the measure induced by the pseudo-Riemannian metric.

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Theorem 1. If F1, . . . , Fk are the faces of P0, Ni is the outer unit normalvector field of P0 along Fi and the vector fields Xi ∈ Γ(TM |Fi

) are compatiblewith the variation, then we have

V ′(0) =k∑

i=1

∫Fi

νidσi =k∑

i=1

∫Fi

εi{X,Ni}dσi. (8)

Proof. We can use the standard idea to prove this theorem. If M is orientable,then we can take the volume form ω of M and apply Stokes’ theorem forthe pull-back π∗ω of ω by the canonical projection π : M × I → M onthe truncated trace Pt

0 = P ∩∗ H0 ∩∗ Ht ⊂ M × I of the variation. Asd(π∗ω) = π∗(dω) = 0, the integral of π∗ω on the boundary of Pt

0 is 0. Pt0

has k lateral faces F1, . . . , Fk and two additional faces Pt×{t} and P0×{0}.Computing the integrals on the lateral faces of Pt

0 by the Fubini theorem weobtain

V (t)− V (0) =

∫ t

0

(k∑

i=1

∫Fi,t

εi{X,Ni}dσi

)dt. (9)

For every 0 ≤ i ≤ k, Fi,t is a 2-transversal CSG solid in ∂Pi,t and there is anisotopy Φ : ∂Pi×I → M for which Φt(∂Pi) = ∂Pi,t. Pulling back the domainof the inner integral on the right hand side of (9) by Φt onto Φ−1

t (Fi,t), weobtain an integral of type (7). Therefore by Proposition 4, the integral inthe bracket in (9) is a continuous function of t, thus, differentiating (9) gives(8).

If M is not orientable, then take its 2-fold orientable covering π : M → Mand apply the above arguments to the lift π−1(Pt) of the variation.

3.2 Some useful formulae

Lemma 1. Let M be an (m + 1)-dimensional Einstein manifold, Σ be asmooth hypersurface in M with a fixed unit normal vector field N. Denoteby H : Σ → R the trace of the Weingarten map and by II the second fun-damental form of Σ. Then for any tangential vector field X along Σ, wehave

XH + δII(X) = 0, (10)

where δII denotes the divergence of the second fundamental form.

Proof. To prove the identity at a given point of Σ choose a tangential or-thonormal frame E1, . . . , Em in an open neighborhood U of that point and

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restrict attention to U . Since M is Einstein, the Ricci endomorphism Ricand the curvature tensor R of M satisfies

Ric(X) =m∑

i=1

ε(Ei)R(X, Ei)Ei + ε(N)R(X,N)N =s

m + 1X,

where s is the scalar curvature of M . Taking the inner product of thisequation with N we obtain

0 =m∑

i=1

ε(Ei){R(X, Ei)Ei,N}. (11)

According to the Codazzi-Mainardi equation

{R(X, Ei)Ei,N} = ε(N)(∇XII(Ei, Ei)− ∇EiII(X, Ei)), (12)

where ∇ is the Levi-Civita connection of Σ. Plugging (12) into (11) we get

0 =m∑

i=1

ε(Ei)(∇XII(Ei, Ei)− ∇EiII(X, Ei)) = {I, ∇XII}+ δII(X). (13)

Since H = {I, II} and ∇XI = 0, we also have

XH = {∇XI, II}+ {I, ∇XII} = {I, ∇XII}. (14)

Equation (10) is an obvious consequence of (13) and (14).

Lemma 2. If Σ is a pseudo-Riemannian smooth hypersurface in an (m +1)-dimensional pseudo-Riemannian manifold (M, {, }), L is the Weingartenmap with respect to a fixed unit normal field N, I and II are the first andsecond fundamental forms , then for any tangent vector field X on Σ we have

δII(X) =1

2{LXI, II} − div L(X),

where LX is the Lie derivative with respect to X.

Proof. Choose a local orthonormal frame E1, . . . , Em on Σ. Then

δII(X) = −m∑

i=1

ε(Ei)∇EiII(X, Ei),

13

and

LXI(Ei, Ej) = −{LXEi, Ej} − {Ei,LXEj} = −{[X, Ei], Ej} − {Ei, [XEj]}= {−∇XEi + ∇Ei

X,Ej}+ {Ei,−∇XEj + ∇EjX}

= −X{Ei, Ej}+ {∇EiX, Ej}+ {Ei, ∇Ej

X}= {∇Ei

X, Ej}+ {Ei, ∇EjX},

from which

1

2{LXI, II} =

m∑i,j=1

ε(Ei)ε(Ej){∇EiX,Ej}II(Ei, Ej)

=m∑

i=1

ε(Ei)II(Ei, ∇EiX).

Finally

div L(X) =m∑

i=1

ε(Ei){∇Ei(L(X)), Ei}

=m∑

i=1

ε(Ei)(Ei(II(X, Ei))− II(X, ∇EiEi))

=m∑

i=1

ε(Ei)(∇EiII(X, Ei)) + II(∇Ei

X, Ei)).

Combining these formulae we obtain the statement.

Lemmas 1 and 2 yield the following.

Corollary 1. For pseudo-Riemannian hypersurfaces lying in pseudo-Rie-mannian Einstein manifolds we have

0 = XH +1

2{LXI, II} − div L(X) for any X ∈ Γ(TΣ).

Let Σ be a smooth pseudo-Riemannian hypersurface in a pseudo-Rie-mannian manifold M , N ∈ Γ(TM |Σ) be a unit normal vector field alongΣ. Consider an isotopy Φt : Σ → M , t ∈ (−ε, ε) and the induced vectorfield X ∈ Γ(TM |Σ), Xp = d

dtΦt(p)

∣∣t=0

. Denote by It, IIt and Ht = {It, IIt}the pull-backs of the first and second fundamental forms and the Minkowskicurvature of Φt(Σ) onto Σ by Φt. Denote by I = I0, II = II0 and III the

14

first, second and third fundamental forms of Σ, and let the symbols I ′, II ′

and H ′ stand for dIt

dt(0), dIIt

dt(0) and dHt

dt(0) respectively. Let X = XT + νN

be the decomposition of X into components tangential and normal to Σ.Choose a point p ∈ Σ and let γ : t 7→ Φt(p) be the curve drawn by p

during the isotopy. Extend Np to a smooth vector field Nγ along γ for whichNγ(t) is a unit normal of Φt(Σ) at γ(t).

Lemma 3. Using the notation introduced above we have

I ′ = −2νII + LXT I, (15)

∇tNγ(0) = −ε(Np) grad ν − Lp(XTp ), (16)

II ′ = ε(N)∇2ν + ν(R(N, ., .,N)− III) + LXT II, (17)

H ′ = ν{II, II} − ε(N)∆ν + ε(N)νr(N) + XT H, (18)

where Lp is the Weingarten map of Σ at p, the gradient of ν is taken in Σ,∇2ν is the Hessian of ν with respect to the Levi-Civita connection, ∆ν =− div(grad ν) = −{I,∇2ν} is the Laplacian of ν, r(N) is the Ricci curvatureof M in the direction N.

Proof. Observe first that (15) and (17) claim the equality of symmetric (0,2)tensor fields, for which it is enough to show the equality of the correspondingquadratic forms. Choose an arbitrary tangent vector Y ∈ TpΣ and a curve η :(−δ, δ) → Σ such that η(0) = p and η′(0) = Y . Consider the parameterizedsurface r : (−ε, ε) × (−δ, δ) → M , r(t, u) = Φt(η(u)) Differentiating theequation It(Y, Y ) = {∂ur(t, 0), ∂ur(t, 0)} at t = 0 we get

I ′(Y, Y ) = 2{∇t∂ur(0, 0), Y } = 2{∇u∂tr(0, 0), Y } = 2{∇Y X, Y }.

Plugging in the decomposition of X, we obtain

I ′(Y, Y ) = 2ν{∇Y N, Y }+ 2{∇Y XT , Y } = −2νII(Y, Y ) + LXT I(Y, Y ).

This proves (15).Extend Nγ to a smooth vector field Nr along r in such a way that Nr(t, u)

is a unit normal of the hypersurface Φt(Σ) at Φt(γ(u)). Since {Nr,Nr} isconstant ±1, ∇tNr(0, 0) = ∇tNγ(0) is tangent to Σ at p. On the other hand,differentiating {Nr, ∂ur} ≡ 0 with respect to t we obtain

{∇tNγ(0), Y } = −{Np,∇t∂ur(0, 0)} = −{Np,∇u∂tr(0, 0)} = −{Np,∇Y X}.

Decomposing X gives

−{Np,∇Y X} = −Y (ν)ε(Np)− II(XTp , Y ) = {−ε(Np) grad ν − Lp(X

Tp ), Y },

15

which yields (16). Differentiating the equation

IIt(Y, Y ) = −{∇uNr(t, 0), ∂ur(t, 0)}

with respect to t at t = 0 we obtain

II ′(Y, Y ) = −{∇t∇uNr(0, 0), Y } − {∇Y N,∇t∂ur(0, 0)}= −{∇u∇tNr(0, 0), Y } −R(Xp, Y,Np, Y )− {∇Y N,∇u∂tr(0, 0)}= ε(Np){∇Y grad ν, Y }+ {∇Y (L(XT )), Y }+ ν(p)R(Np, Y, Y,Np)

− {∇Y N,∇Y X} −R(XTp , Y,Np, Y )

= ε(Np){∇Y grad ν, Y }+ ν(p)(R(Np, Y, Y,Np)− {∇Y N,∇Y N})+ {∇Y (L(XT )), Y } − {∇Y N,∇Y XT} −R(XT

p , Y,Np, Y ).

In the special case when X is tangent to Σ and Φt is the flow generated byX on Σ, the above formula reduces to

LXT II(Y, Y ) = {∇Y (L(XT )), Y } − {∇Y N,∇Y XT} −R(XTp , Y,Np, Y ),

from which

II ′(Y, Y ) = ε(Np){∇Y grad ν, Y }+ ν(p)(R(Np, Y, Y,Np)− {∇Y N,∇Y N})+ LXT II(Y, Y ),

proving (17).Let Gt and Bt be the matrices of It and IIt with respect to a local orthonor-

mal frame over an open subset U of Σ. Then Ht = trG−1t Bt. Differentiating

this equation we obtain

H ′ = tr(G−10 B′0 − G−1

0 G ′0G−10 B0) = {I, II ′} − {I ′, II}

= 2ν{II, II} − ε(N)∆ν + ε(N)νr(N)− ν{I, III}+ XT H

= ν{II, II} − ε(N)∆ν + ε(N)νr(N) + XT H.

Corollary 2. If M is an (m + 1)-dimensional pseudo-Riemannian Einsteinmanifold with sectional curvature s, Σ is a smooth pseudo-Riemannian hy-persurface, X is an infinitesimal variation of Σ as above, then

s

m + 1{X,N} = H ′ +

1

2{I ′, II} − div(ε(N) grad ν + L(XT )).

16

Proof. Using (18), (15) and Corollary 1

s

m + 1{X,N} = ε(N)νr(N) = H ′ − ν{II, II}+ ε(N)∆ν −XT H

= H ′ +1

2{I ′, II}+ ε(N)∆ν − 1

2{LXT I, II} −XT H

= H ′ +1

2{I ′, II} − div(ε(N) grad ν + L(XT )).

3.3 Variation of the dihedral angle

Consider a variation Pt = f ∗(P1,t, . . . , Pk,t) of a pseudo-Riemannian polytopein a pseudo-Riemannian manifold (M, {, }), let Wij be the wall of P = P0

between the i-th and j-th faces. Choose an isotopy Φ : (∂Pi∩ ∂Pj)× I → Msuch that Φt(∂Pi ∩ ∂Pj) = (∂Pi,t ∩ ∂Pj,t) and Φ0 is the inclusion, and letXij ∈ Γ(TM |Wij

) be the initial speed vector field Xij(p) = dΦdt

∣∣t=0

. Definethe signs εi, εj, εij, εji, the functions αij, λ, µ and the vector fields Ni, Nj,nij, nji along Wij as in subsection 2.3. These functions and vector fieldscan be extended to functions and vector fields along the isotopy Φ. Forexample we extend αij to a smooth function αΦ

ij : (∂Pi ∩ ∂Pj) × I → R insuch a way that αΦ

ij(p, 0) ≡ αij and αΦij(p, t) is one of the angles between the

∂Pi,t and ∂Pj,t at the point Φ(p, t). Derivatives of the extended functionsαΦ

ij, λΦ, µΦ and covariant derivative of the extended vector fields NΦi , NΦ

j ,nΦ

ij, nΦji with respect to t at t = 0 will be denoted by Xijα

Φij, Xijλ

Φ, XijµΦ

∇XijNΦ

i , ∇XijNΦ

j , ∇XijnΦ

ij and ∇XijnΦ

ji respectively. It will be seen thatthese derivatives depend only on Xij, so to simplify notation we drop the Φ’sfrom the upper indeces.

Our aim is to find an expression for the derivative Xijαij of the dihedralangle. Differentiating the relation

{Ni,Nj} = −εiεjiλ

and making use ofXijλ = −εiεij(Xijαij)µ

we obtain

εijεji(Xijαij)µ = {∇XijNi,Nj}+ {Ni,∇Xij

Nj}. (19)

A computation similar to the one used in the proof of (16) shows that

∇XijNi = −εi grad νi − Li(Xij − νiNi),

17

where νi = εi{Xij,Ni}, Li is the Weingarten map of the hypersurface ∂Pi.Taking inner product with (4) and using {∇Xij

Ni,Ni} = 0 this yields

{∇XijNi,Nj} = −εiεjiµ{εi grad νi + Li(Xij − νiNi),nij} (20)

Plugging (20) into (19) and dividing by µ (6= 0 by 2-transversality) we get

Xijαij =− εi{εi grad νi + Li(Xij − νiNi), εijnij}− εj{εj grad νj + Lj(Xij − νjNj), εjinji}.

(21)

3.4 Schlafli formula for polytopes

Theorem 2. We have the following formula for the variation of the volume ofa polytope in an (m + 1)-dimensional pseudo-Riemannian Einstein manifoldwith scalar curvature s

s

m + 1V ′(0) =

k∑i=1

εi

∫Fi

(H ′i +

1

2{I ′i, IIi})dσi

+∑

1≤i<j≤k

∫Wij

(Xijαij)dσij

+k∑

i=1

k∑j=1j 6=i

∫Wij

εiεij{Li(Xij −Xi),nij}dσij.

(22)

Proof. By Theorem 1 and Corollary 2, we have

s

m + 1V ′(0) =

k∑i=1

∫Fi

εis

m + 1{X,Ni}dσi

=k∑

i=1

εi

∫Fi

(H ′i +

1

2{I ′i, IIi})dσi −

k∑i=1

εi

∫Fi

div(εi grad νi + Li(XTi ))dσi.

The last family of integrals can be computed by the divergence theorem∫Fi

div(εi grad νi + Li(XTi ))dσi =

∑1≤j≤k

j 6=i

∫Wij

{εi grad νi + Li(XTi ), εijnij}dσij

and thus the theorem follows from (21) and the equation XTi = Xi−νiNi.

18

4 Application to the Kneser-Poulsen conjec-

ture

Let M be the (m+1)-dimensional hyperbolic, Euclidean or spherical space ofconstant sectional curvature K = s/(m2+m). Complete connected umbilicalhypersurfaces in M will be called ∗-spheres. The term ∗-ball will be used forany ball if K > 0 and for any convex regular domain bounded by a ∗-sphereif K ≤ 0. In this section we study the consequences of the Schlafli typeformula (22) for polytopes made of ∗-balls in M .

Let P = f ∗(B1, . . . , Bk) be a polytope made of ∗-balls in M , such that theboundary of Bi is a ∗-sphere with constant normal curvature κi. Consider avariation Pt of P , obtained by moving the primitives Bi rigidly. In this casewe can choose Killing fields for the infinitesimal variations Xi. Doing so, theintegrals

∫Fi

(H ′i +

12{I ′i, IIi})dσi in (22) will all vanish. The intersection angle

of ∗-spheres is constant along the intersection, so the funtions αij dependonly on t and the derivative α′

ij(0) = Xijαij does not depend on the actualchoice of Xij. By umbilicity, the Weingarten map Li of ∂Bi is simply amultiplication by κi, thus, formula (22) reduces to the following form

s

m + 1V ′(0) =

∑1≤i<j≤k

∫Wij

α′ij(0)σij(Wij)

+k∑

i=1

k∑j=1j 6=i

∫Wij

κi{(Xij −Xi),nij}dσij.

Since Xi is a Killing field, and Xij is a compatible infinitesimal variation

of the wall Wij,∑k

j=1j 6=i

∫Wij{(Xij − Xi),nij}dσij is the derivative of the m-

dimensional volume of the i-th face at t = 0 (cf. Theorem 1). Thus weobtain the following special case of Theorem 2.

Theorem 3. For a polytope built of ∗-balls in M , we have the followingvariational formula

d

dt

(s

m + 1V −

k∑i=1

κiσi(Fi)

)∣∣∣∣∣t=0

=∑

1≤i<j≤k

α′ij(0)σij(Wij). (23)

Corollary 3. If a polytope P = f ∗(B1, . . . , Bk) is varied within the class ofpolytopes by moving the ∗-balls Bi smoothly and rigidly in such a way that theinner dihedral angles αij of the polytope weakly decrease, then the quantity(sV/(m + 1)−

∑i κiσi(Fi)) does not increase.

19

For simplicity call CSG objects made of ordinary balls flowers. We aregoing to extend Corollary 3 for piecewise analytic variations of flowers.

The boundary of a flower is typically not a smooth submanifold of M , butthe singular points always form a negligible set (see [15]). At smooth pointsof the boundary, the boundary is umbilical and its principal curvature κ withrespect to the outer unit normal is well defined. The function which assigns toa flower P the number sV (P )/(m+1)−

∫∂P

κdσ is a natural extension of thefunction “sV/(m+1)−

∑i κiσi(Fi)”, which was defined only for polytopes. It

is not difficult to see that this function varies continuously under a variationof the flower.

For ordinary intersecting balls Bi = B(Pi, ri) and Bj = B(Pj, rj), theinner dihedral angle α(Bi, Bj) of the union Bi ∪Bj is an increasing functionof the distance d(Pi, Pj) between the centers. Taking a regularized Booleanexpression f ∗ in which each variable is used exactly once, one can define asign εf∗

ij depending only on f ∗ such that for any 3-transversal family of ballsthe inner dihedral angle αij of the poytope f ∗(B1, . . . , Bk) coincides with

εf∗

ij α(Bi, Bj) plus a constant multiple of π. Therefore, the inner dihedralangle αij of a 3-transversal flower built of balls with fixed radii is an increasing

function of the signed distance εf∗

ij d(Pi, Pj) of the centers.

Theorem 4. If a flower P = f ∗(B1, . . . , Bk) is varied by moving the ballsBi = B(Pi, ri) by piecewise analytic rigid motions in such a way that thesigned distances εf∗

ij dij(t) = εf∗

ij d(Pi,t, Pj,t) between the moving centers weaklydecrease, then the quantity sV (Pt)/(m + 1)−

∫∂Pt

κdσ does not increase.

Proof. Since the quantity sV (Pt)/(m+1)−∫

∂Ptκdσ varies continuously dur-

ing the variation, it is enough to show monotonicity on intervals where thevariation is analytic. Configurations of balls not satisfying the 3-transversalitycondition can be characterized by the vanishing of some analytic functions soby the unicity theorem, if Pt is a polytope for at least one moment of time,then it will break the 3-transversality condition only at a finite number ofmoments. These singular moments cut the time interval into a finite num-ber of intervals. On each subinterval, sV (Pt)/(m + 1) −

∫∂Pt

κdσ does notincrease, so we are done by continuity.

If the configuration of the balls is singular during the whole variation,then we can perturb the radii. The set S of vectors δ = (δ1, . . . , δk) for whichri + δi ≥ 0 and the balls Bδ

i,t = B(Pi, ri + δi) satisfy the 3-transversalitycondition is nowhere dense in Rk. For δ /∈ S, we know the monotonicityof “sV (P δ

t )/(m + 1) −∫

∂P δtκdσ” for the variation P δ

t = f ∗(Bδ1,t, . . . , B

δk,t).

Taking the limit as δ /∈ S tends to 0 we get the required monotonicity in thegeneral case.

20

Theorem 4 has the same flavour as the Kneser-Poulsen conjecture. It willturn out below that the connection is more organic than just a superficialsimilarity.

Let N be an (m − 1)-dimensional complete totally geodesic subspace ofM . In the spherical case, let N∗ be the polar circle of N and set N∗ = ∅ forK ≤ 0. The group G of orientation preserving isometries of M leaving eachpoint of N fixed consists of rotations of M about N , hence G is isomorphicto the circle group S1.

Theorem 5. Suppose that a the set of singular points of the boundary of aG-invariant CSG object P in M is negligible and that the boundary hyper-surfaces of the primitives of P intersect N transversally. If a non-singularboundary point p ∈ ∂P is not in N , then denote by kG(p) the normal curva-ture of ∂P in the direction tangent to the circle Gp, relative to the exteriorunit normal vector field of P . Then the following identity holds

s

m + 1Vm+1(P )−

∫∂P

kGdσ = 2πVm−1(P ∩N), (24)

where Vm+1, Vm−1 and σ are the volume measures induced by the Riemannianmetric on M , N and the smooth part of ∂P respectively.

Proof. First we recall some formulas concerning tubular hypersurfaces aroundN . We refer to [10] for details. Tubular hypersurfaces around N are obtainedas the level sets of the function ρ : M → R, which assigns to a point in Mits geodesic distance from N .

By the symmetry properties of the tubular hypersurfaces around N , thetubular hypersurface of radius a (a < π/(2

√K) in the spherical case) has

two principal curvatures κ1(a) and κ2(a) with multiplicities 1 and (m − 1)respectively. These principal curvature functions satisfy the Riccati differen-tial equation κ′∗ = κ2

∗ + K with initial condition κ1(0) = −∞ and κ2(0) = 0,from which κ1(a) = −

√K cot(

√Ka) and κ2(a) =

√K tan(

√Ka). In partic-

ular, the trace of the shape operator of the tubular hypersurface of radiusa around N is

√K((m − 1) tan(

√Ka) − cot(

√Ka)). Observe that κ1(a) is

the principal curvature of a sphere of radius a with respect to the outer unitnormal vector field.

The normal bundle of N is trivial and carries a natural Riemannian prod-uct metric, so it can be identified with N ×R2. The exponential map of thenormal bundle gives a diffeomorphism between N ×R2 in the non-positivelycurved cases and between the tube of radius π/(2

√K) around the zero sec-

tion of the bundle and M \ N∗ in the spherical case. The pull-back of thevolume measure of M onto N × R2 by the exponential map is the volume

21

measure of N ×R2 multiplied by a smooth function of the form ϑ ◦ ρ, whereρ : N ×R2 → R is the pull-back of the distance function ρ, i.e. ρ(p,x) = |x|,ϑ is a smooth function whose logarithmic derivative is

ϑ′

ϑ(a) = −(a−1 +

√K((m− 1) tan(

√Ka)− cot(

√Ka))). (25)

Integrating (25) taking care of the initial condition ϑ(0) = 1 we obtain

ϑ(a) =sin(

√Ka) cosm−1(

√Ka)√

Ka. (26)

Let X be the unit vector field on M \ (N ∪N∗) whose restriction onto atubular hypersurface around N is the unit normal vector field of the hyper-surface pointing away from N .

The vector field Y = (κ1 ◦ ρ)X has a singularity along N , but extendssmoothly to N∗ in the spherical case. By symmetry, the divergence of Ydepends only on the distance of a point from N , thus div Y = g ◦ ρ for asuitable function g. To determine g explicitly, take the solid tubes Aa andAb of radii a > b > 0 around an arbitrary compact regular domain A in Nand apply the divergence theorem for the vector field Y over Aa \∗ Ab. Thendividing by the volume of A we get

2π

∫ a

b

g(t)sin(

√Kt) cosm−1(

√Kt)√

Kdt = 2π[κ1(t)tϑ(t)]ab ,

from which div Y = g ◦ ρ turns out to be constant Km = s/(m + 1).Let ε be a positive number and consider the solid tube Nε around N . If

ε is sufficiently small, then the boundary of Nε intersects the smooth part ofthe boundary of P almost orthogonally and therefore transversally and theintersection ∂Nε ∩ (∂P ) is covered by the 2ε-neighborhood of N ∩ ∂P . Inthat case we can apply the divergence theorem for the domain P \∗ Nε:∫

P\∗Nε

s

m + 1dVm+1 =

∫∂P\Nε

κ1 ◦ ρ{X,N}dσ − κ1(ε)σ(∂Nε ∩ P ), (27)

where N is the outer unit normal vector field on the smooth part of theboundary of P , {, } is the Riemannian metric of M , σ is the m dimensionalvolume measure induced by the Riemannian metric on smooth hypersurfacesof M .

Study the limit of the components of the equation (27) as ε tends to 0.The limit of the left hand side is just sVm+1(P )/(m + 1). If p ∈ ∂P is asmooth boundary point of P , then ∂P contains the circle Gp. The curvature

22

of this circle is κ1(ρ(p)) provided that the orientations are chosen in sucha way that X is the second Frenet vector field of the circle. Accordingto Meusnier’s theorem, the curvature of the circle is related to the normalcurvature of the hypersurface in the direction tangent to this circle by theequation κ1 ◦ρ{X,N} = kG. This implies that the integral on the right handside of (27) converges to

∫∂P

kGdσ.Consider the intersection P∩N and the 2ε-neighborhood U2ε of its relative

boundary in N . The orthogonal projection of ∂Nε ∩ P is contained in (P ∩N)∪U2ε and it contains (P ∩N) \U2ε, therefore κ1(ε)σ(∂Nε∩P ) is between2πκ1(ε)εϑ(ε)Vm−1((P ∩N)∪U2ε) and 2πκ1(ε)εϑ(ε)Vm−1((P ∩N)\U2ε). Since

limε→0

Vm−1((P ∩N) ∪ U2ε) = limε→0

Vm−1((P ∩N) \ U2ε) = Vm−1(P ∩N)

and limε→0−κ1(ε)εϑ(ε) = 1, the limit of (27) as ε tends to 0 yields (24).

Corollary 4. Suppose that P = f ∗(B1, . . . , Bk) is a CSG object in M madeof G-invariant ∗-balls. Denote by κ the principal curvature function on thesmooth part of the boundary ∂P and by σ the m-dimensional volume measureon ∂P . Then we have

s

m + 1Vm+1(P )−

∫∂P

κdσ = 2πVm−1(f∗(B1 ∩N, . . . , Bk ∩N)).

Combining Theorem 4 and Corollary 4 we obtain

Theorem 6. Let f ∗ be a Boolean expression as usual, P1, . . . , Pk andQ1, . . . , Qk be points in the (m − 1)-dimensional subspace N . If there existpiecewise analytic curves γi : [0, 1] → M connecting the points γi(0) = Pi tothe points γi(1) = Qi in such a way that the signed distances εf∗

ij d(γi(t), γj(t))weakly decrease as t increase, then for any choice of the radii, we have thefollowing inequality between the volumes of the (m− 1)-dimensional flowersbuilt from the balls BN

i = B(Pi, ri) ∩N and BNi = B(Qi, ri) ∩N :

Vm−1(f∗(BN

1 , . . . , BNk )) ≤ Vm−1(f

∗(BN1 , . . . , BN

k )).

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23

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[11] Y. Gordon and M. Meyer, On the Volume of Unions and Intersections ofBalls in the Euclidean Space, Oper. Theory Adv. Appl. 77, Birkhauser,Basel, 1995, 91–100.

[12] M. Gromov, Monotonicity of the Volume of Intersection of Balls, (Gafa85-86), Lecture Notes in Math. 1267, Springer, New York, 1987, 1–4.

[13] H. Hadwiger, Ungeloste Probleme No. 11, Elem. Math., 11 (1956), 60–61.

[14] M. Kneser, Einige Bemerkungen uber das Minkowskische Flachenmass,Arch. Math. 6 (1955), 382–390.

[15] S. Lang, Fundamentals Of Differential Geometry, Graduate Texts InMathematics 191, Springer Verlag New York, 1999.

[16] E.T. Poulsen, Problem 10, Math. Scand. 2 (1954), 346.

[17] I. Rivin and J.-M. Schlenker, The Schlafli formula in Einstein manifoldswith boundary, Electron. Res. Announc. Amer. Math. Soc. , 5 (1999)18–23.

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[18] R. Souam, The Schlafli formula for polyhedra and piecewise smoothhypersurfaces, Differential Geom. Appl., 20 (2004), no. 1, 31–45.

[19] F.W. Warner, Foundation of Differentiable Manifolds and Lie Groups,Graduate Texts in Mathematics 94, Springer Verlag New York, 1983.

Author’s address: Dept. of Geometry, Eotvos University, Budapest, P.O.B. 120, H-1518Hungarye-mail: csikos@cs.elte.hu

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