# Analogues of Alexandrov's and Stoker's theorems for ball-polyhedra

**ABSTRACT** The rigidity theorems of Alexandrov (1950) and Stoker (1968) are classical

results in the theory of convex polyhedra. In this paper we prove analogues of

them for normal (resp., standard) ball-polyhedra. Here, a ball-polyhedron means

an intersection of finitely many congruent balls in Euclidean 3-space.

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**ABSTRACT:**This is an exciting book aimed at graduate students and researchers as it contains a multitude of open problems after leading the reader to the research frontier. From the publisher: “This monograph gives a short introduction to parts of modern discrete geometry, in addition to leading the reader to the frontiers of geometric research on sphere arrangements. The readership is aimed at advanced undergraduate and early graduate students, as well as interested researchers. It contains 30 open research problems ideal for graduate students and researchers in mathematics and computer science. Additionally, this book may be considered ideal for a one-semester advanced undergraduate or graduate level course. The core of this book is based on three lectures given by the author at the Fields Institute during the thematic program on discrete geometry and applications and contains four basic topics. The first two deal with active areas that have been outstanding from the birth of discrete geometry, namely dense sphere packings and tilings. Sphere packings and tilings have a very strong connection to number theory, coding, groups, and mathematical programming. Extending the tradition of studying packings of spheres is the investigation of the monotonicity of volume under contractions of arbitrary arrangements of spheres. The third major topic can be found under the sections on ball-polyhedra that study the possibility of extending the theory of convex polytopes to the family of intersections of congruent balls. This section of the text is connected in many ways to the above-mentioned major topics as well as to some other important research areas such as that on coverings by planks (with close ties to geometric analysis). The fourth basic topic is discussed under covering balls by cylinders.”

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arXiv:1201.3656v1 [math.MG] 17 Jan 2012

Analogues of Alexandrov’s and Stoker’s

theorems for ball-polyhedra∗

K´ aroly Bezdek†

January 19, 2012

Abstract

The rigidity theorems of Alexandrov (1950) and Stoker (1968) are

classical results in the theory of convex polyhedra.

we prove analogues of them for (standard as well as normal) ball-

polyhedra. Here, a ball-polyhedron means an intersection of finitely

many congruent balls in Euclidean 3-space.

In this paper

1Introduction

First, we recall the notation of ball-polyhedra, the central object of study

for this paper. Let E3denote the 3-dimensional Euclidean space. As in

[4] and [5] a ball-polyhedron is the intersection with non-empty interior of

finitely many closed congruent balls in E3. In fact, one may assume that the

closed congruent 3-dimensional balls in question are of unit radius; that is,

they are unit balls of E3. Also, it is natural to assume that removing any

of the unit balls defining the intersection in question yields the intersection

of the remaining unit balls becoming a larger set. (Equivalently, using the

terminology introduced in [5], whenever we take a ball-polyhedron we always

∗Keywords: Cauchy’s rigidity theorem, Alexandrov’s theorem, Stoker’s theorem, stan-

dard ball-polyhedron, normal ball-polyhedron, analogues of Alexandrov’s and Stoker’s

theorems for ball-polyhedra. 2010 Mathematics Subject Classification: 52C25, 52B10,

and 52A30.

†Partially supported by a Natural Sciences and Engineering Research Council of

Canada Discovery Grant.

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assume that it is generated by a reduced family of unit balls.) Furthermore,

following [4] and [5] one can represent the boundary of a ball-polyhedron

in E3as the union of vertices, edges, and faces defined in a rather natural

way as follows. A boundary point is called a vertex if it belongs to at least

three of the closed unit balls defining the ball-polyhedron. A face of the

ball-polyhedron is the intersection of one of the generating closed unit balls

with the boundary of the ball-polyhedron. Finally, if the intersection of two

faces is non-empty, then it is the union of (possibly degenerate) circular arcs.

The non-degenerate arcs are called edges of the ball-polyhedron. Obviously,

if a ball-polyhedron in E3is generated by at least three unit balls, then it

possesses vertices, edges, and faces. Clearly, the vertices, edges and faces of a

ball-polyhedron (including the empty set and the ball-polyhedron itself) are

partially ordered by inclusion forming the vertex-edge-face structure of the

given ball-polyhedron. It was an important observation of [4] as well as of

[5] that the vertex-edge-face structure of a ball-polyhedron is not necessarily

a lattice (i.e., a partially ordered set (also called a poset) in which any two

elements have a unique supremum (the elements’ least upper bound; called

their join) and an infimum (greatest lower bound; called their meet)). Thus,

it is natural to define the following fundamental family of ball-polyhedra,

introduced in [5] under the name standard ball-polyhedra and investigated in

[4] as well without having a particular name for it. Here a ball-polyhedron

in E3is called a standard ball-polyhedron if its vertex-edge-face structure is

a lattice (with respect to containment). In this case, we simply call the

vertex-edge-face structure in question the face lattice of the standard ball-

polyhedron. This definition implies among others that any standard ball-

polyhedron of E3is generated by at least four unit balls.

Second, we state our new results on ball-polyhedra together with some

well-known theorems on convex polyhedra. In fact, those classical theorems

on convex polyhedra have motivated our work on ball-polyhedra a great deal

furthermore, their proofs form the bases of our proofs in this paper. The

details are as follows. One of the best known results on convex polyhedra is

Cauchy’s celebrated rigidity theorem [6]. (For a recent account on Cauchy’s

theorem see Chapter 11 of the mathematical bestseller [1] as well as Theorem

26.6 and the discussion followed in the elegant book [7].) Cauchy’s theorem

is often quoted as follows: If two convex polyhedra P and P′in E3are com-

binatorially equivalent with the corresponding faces being congruent, then

P is congruent to P′. It is immediate to note that the analogue of Cauchy’s

theorem for ball-polyhedra is a rather obvious statement and so, we do not

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discuss that here. Next, it is natural to recall Alexandrov’s theorem [3] in

particular, because it implies Cauchy’s theorem (see also Theorem 26.8 and

the discussion followed in [7]): if P and P′are combinatorially equivalent

convex polyhedra with equal corresponding face angles in E3, then P and P′

have equal corresponding inner dihedral angles. Somewhat surprisingly, the

analogue of Alexandrov’s theorem for ball-polyhedra is not trivial. Still, one

can prove it following the ideas of the original proof of Alexandrov’s theorem

[3]. This was published in [4] (see Claim 5.1 and the discussion followed).

Here, we just state the theorem in question for later use and in order to

do so, we need to recall some additional terminology. To each edge of a

ball-polyhedron in E3we can assign an inner dihedral angle. Namely, take

any point p in the relative interior of the edge and take the two unit balls

that contain the two faces of the ball-polyhedron meeting along that edge.

Now, the inner dihedral angle along this edge is the angular measure of the

intersection of the two half-spaces supporting the two unit balls at p. The

angle in question is obviously independent of the choice of p. Moreover, at

each vertex of a face of a ball-polyhedron there is a face angle which is the

angular measure of the convex angle formed by the two tangent half-lines of

the two edges meeting at the given vertex. Finally, we say that the standard

ball-polyhedra P and P′in E3are combinatorially equivalent if there is an

inclusion (i.e., partial order) preserving bijection between the face lattices of

P and P′. Thus, [4] proves the following analogue of Alexandrov’s theorem

for standard ball-polyhedra: If P and P′are two combinatorially equivalent

standard ball-polyhedra with equal corresponding face angles in E3, then P

and P′have equal corresponding inner dihedral angles.

An important close relative of Cauchy’s rigidity theorem is Stoker’s the-

orem [10] (see also Theorem 26.9 and the discussion followed in [7]): if P

and P′are two combinatorially equivalent convex polyhedra with equal cor-

responding edge lengths and inner dihedral angles in E3, then P and P′are

congruent. As it turns out, using the ideas of original proof ([10]) of Stoker’s

theorem, one can give a proof of the following analogue of Stoker’s theorem

for standard ball-polyhedra.

Theorem 1.1 If P and P′are two combinatorially equivalent standard ball-

polyhedra with equal corresponding edge lengths and inner dihedral angles in

E3, then P and P′are congruent.

Based on the above mentioned analogue of Alexandrov’s theorem for stan-

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dard ball-polyhedra, Theorem 1.1 implies the following statement in straight-

forward way.

Corollary 1.2 If P and P′are two combinatorially equivalent standard ball-

polyhedra with equal corresponding edge lengths and face angles in E3, then

P and P′are congruent.

In order to strengthen the above mentioned analogue of Alexandrov’s

theorem for standard ball-polyhedra, we recall the following notion from [4].

We say that the standard ball-polyhedron P in E3is globally rigid with respect

to its face angles within the family of standard ball-polyhedra if the following

holds. If P′is another standard ball-polyhedron in E3whose face lattice is

combinatorially equivalent to that of P and whose face angles are equal to

the corresponding face angles of P, then P′is congruent to P. Furthermore,

a ball-polyhedron of E3is called triangulated if all its faces are bounded by

three edges. It is not hard to see that any triangulated ball-polyhedron is,

in fact, a standard one. Now, recall the following theorem proved in [4] (see

Theorem 0.2): if P is a triangulated ball-polyhedron in E3, then P is globally

rigid with respect to its face angles. This raises the following question.

Problem 1.3 Prove or disprove that every standard ball-polyhedron of E3

is globally rigid with respect to its face angles within the family of standard

ball-polyhedra.

Actually, the restriction that only standard ball-polyhedra are considered

in Problem 1.3 is a rather natural one. Namely, if Q is a non-standard ball-

polyhedron, then Q possesses a pair of faces whose intersection consists of at

least two connected components. However, such a ball-polyhedron is always

flexible (and so, it is not globally rigid) as shown in Section 4 of [4].

In this paper we give a positive answer to Problem 1.3 within the following

subfamily of standard ball-polyhedra. In order to define the new family of

ball-polyhedra in an elementary way, we first take a ball-polyhedron P in

E3and then label the union of its generating unit balls by P∪and call it

the flower-polyhedron generated by P. Next, we say that a sphere of E3is

a circumscribed sphere of the flower-polyhedron P∪if it contains P∪(i.e.,

bounds a closed ball containing P∪) and touches some of the unit balls of P∪

such that there is no other sphere of E3touching the same collection of unit

balls of P∪and contaning P∪. Finally, we call P a normal ball-polyhedron if

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the radius of every circumscribed sphere of the flower-polyhedron P∪is less

than 2. In the proof of the following theorem we show that every normal

ball-polyhedron is in fact, a standard ball-polyhedron. (However, there are

standard ball-polyhedra that are not normal ball-polyhedra.) We have the

following analogue of Alexandrov’s theorem for normal ball-polyhedra.

Theorem 1.4 Every normal ball-polyhedron of E3is globally rigid with re-

spect to its face angles within the family of normal ball-polyhedra.

In the rest of the paper we prove the theorems stated.

2Proof of Theorem 1.1

We follow the ideas of the original proof of Stoker’s theorem [10] (see also

the proof of Theorem 26.9 in [7]) with properly adjusting that to the family

of standard ball-polyhedra. The details are as follows.

First, we need to introduce some basic notation and make some simple

observations. In what follows x stands for the notation of a point as well as

of its position vector in E3with o denoting the origin of E3. Moreover, ?·,·?

denotes the standard inner product in E3and so, the corresponding standard

norm is labelled by ? · ? satisfying ?x? =

radius (or simply the unit ball) centered at x is denoted by B[x] := {y ∈

E3| ?x − y? ≤ 1} and its boundary bd(B[x]) := {y ∈ E3| ?x − y? = 1},

the unit sphere with center x, is labelled by S(x) := bd(B[x]). Let P :=

∩f

{B[xk] | 1 ≤ k ≤ f} of f ≥ 4 unit balls. Here, each unit ball B[xk] gives rise

to a face of P namely, to Fk:= S(xk) ∩ bd(P) for 1 ≤ k ≤ f. Clearly, as

P is a standard ball-polyhedron, each edge of P is of the form Fk1∩ Fk2for

properly chosen 1 ≤ k1,k2≤ f and therefore it can be labelled accordingly

with E{k1,k2}. Furthermore, let {E{i,k}| i ∈ Ik⊂ {1,2,...,f}} be the family

of the edges of Fk. Moreover, let {vj| 1 ≤ j ≤ v} denote the vertices of P.

In particular, let the set of the vertices of Fkbe {vj| j ∈ Jk⊂ {1,2,...,v}}.

Next, let α{k1,k2}(resp., βj,k) denote the inner dihedral angle along the edge

E{k1,k2} of P (resp., the face angle at the vertex vj of the face Fk of P).

Finally, let Ck[z,γ] := {y ∈ S(xk) | ?z − xk,y − xk? ≥ cosγ} denote the

closed spherical cap lying on S(xk) and having angular radius 0 < γ ≤ π

??x,x?. The closed ball of unit

k=1B[xk] be a standard ball-polyhedron generated by the reduced family

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Page 6

with center z ∈ S(xk). Then it is rather easy to show that

Fk=

?

i∈Ik

Ck

?

zi,k,α{i,k}

2

?

,(1)

where zi,k:= xk+

Fkis a spherically convex subset of S(xk) (meaning that with any two points

of Fkthe geodesic arc of S(xk) connecting them lies in Fk). Furthermore, (1)

yields that the edges {E{i,k}| i ∈ Ik} of Fkare circular arcs of Euclidean radii

{sin

2

| i ∈ Ik}. Now, let the tangent cone Tvjof P at the vertex vjbe

defined by Tvj:= cl(vj+ pos{y − vj| y ∈ P}), where cl(·) (resp., pos{·})

stands for the closure (resp., positive hull) of the corresponding set. Then it

is natural to define the (outer) normal cone T∗

T∗

as well as T∗

this, it is immediate to define the vertex figure Tvj:= Tvj∩ S(vj) as well

as the normal image T∗

straightforward to make the following two observations. The vertex figure

Tvjof P at vj is a spherically convex polygon of S(vj) with side lengths

(resp., angles) equal to

1

?xi−xk?(xi− xk). As

α{i,k}

2

<π

2therefore (1) implies that

α{i,k}

vjof P at the vertex vj via

vj:= vj+ {y ∈ E3| ?y − vj,z − vj? ≤ 0 for all z ∈ Tvj}. Clearly, Tvj

vjare convex cones of E3with vjas a common apex. Based on

vj:= T∗

vj∩ S(vj) of P at the vertex vj. Now, it is

{βj,k| vj∈ Fk} (resp., {α{k1,k2}| vj∈ E{k1,k2}})(2)

The normal image T∗

with side lengths (resp., angles) equal to

vjof P at vj is a spherically convex polygon of S(vj)

{π − α{k1,k2}| vj∈ E{k1,k2}} (resp., {π − βj,k| vj∈ Fk}) (3)

Having discussed all this, we are ready to take the standard ball-polyhedron

P′:= ∩f

of the above introduced notations for P′are as follows: {F′

bd(P′) | 1 ≤ k ≤ f}; {E′

α′

v′

with z′∈ S(x′

dihedral angles, i.e., α{k1,k2}= α′

follows:

F′

?

i∈Ik

k=1B[x′

k] that is combinatorially equivalent to P. The analogues

k:= S(x′

j| j ∈ Jk};

k,y′−x′

k) ∩

{i,k}| i ∈ Ik}; {v′

j; and C′

k),0 < γ ≤ π. By assumption, P and P′have equal inner

{k1,k2}. Thus, the analogue of (1) reads as

j| 1 ≤ j ≤ v}; {v′

k) | ?z′−x′

{k1,k2}; β′

j,k; Tv′

j; T∗

k[z′,γ] := {y′∈ S(x′

k? ≥ cosγ}

k=C′

k

?

z′

i,k,α{i,k}

2

?

,(4)

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Page 7

where z′

P′at v′

angles) equal to

i,k:= x′

jis a spherically convex polygon of S(v′

k+

1

?x′

i−x′

k?(x′

i− x′

k). In particular, the normal image T∗

j) with side lengths (resp.,

v′

jof

{π − α{k1,k2}| vj∈ E{k1,k2}} (resp., {π − β′

j,k| vj∈ Fk})(5)

Second, we need to recall the two main ideas of the original proof of

Cauchy’s rigidity theorem [6]. The following is called the (spherical) Legend-

re-Cauchy lemma (see Theorem 22.2 and the discussions followed in [7] as

well as [8] for a recent proof and the history of the statement).

Lemma 2.1 Let U and U′be two spherically convex polygons (on an open

hemisphere) of the unit sphere S2:= {y ∈ E3| ?o − y? = 1} with vertices

u1,u2,...,un, and u′

equal corresponding spherical side lengths (or, equivalently, with ?ui+1−ui? =

?u′

are the angular measures of the interior angles ∠ui−1uiui+1and ∠u′

of U and U′at the vertices uiand u′

at least four sign changes in the cyclic sequence γ1− γ′

(in which we simply ignore the zeros) or the cyclic sequence consists of zeros

only.

1,u′

2,...,u′

n(enumerated in some cyclic order) and with

i+1−u′

i? for all 1 ≤ i ≤ n, where un+1:= u1and u′

n+1:= u′

1). If γiand γ′

i−1u′

i

iu′

i+1

ifor 1 ≤ i ≤ n, then either there are

1,γ2− γ′

2,...,γn− γ′

n

The following is called the sign counting lemma (see Lemma 26.5 in [7]

as well as the Proposition in Chapter 10 of [1]). For the purpose of that

statement we recall here that a graph is a pair G := (V,E), where V is the

set of vertices, E is the set of edges, and each edge e ∈ E “connects” two

vertices v,w ∈ V . The graph is called simple if it has no loops (i.e., edges

for which both ends coincides) or parallel edges (that have the same set of

end vertices). In particular, a graph is planar if it can be drawn on S2(or,

equivalently, in E2) without crossing edges. We talk of a plane graph if such

a drawing is already given and fixed.

Lemma 2.2 Suppose that the edges of a simple plane graph are labeled with

0, + and − such that around each vertex either all labels are 0 or there are

at least four sign changes (in the cyclic order of the edges around the vertex).

Then all signs are 0.

Now, we are set for the final approach in proving Theorem 1.1. By as-

sumption P and P′are two combinatorially equivalent standard ball-poly-

hedra with equal corresponding edge lengths and inner dihedral angles in E3.

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Page 8

Thus, α{k1,k2}= α′

families {E{i,k}| i ∈ Ik} and {E′

circular arcs of equal Euclidean radii (namely, sin

(with the latter property holding by assumption). Hence, in order to com-

plete the proof of Theorem 1.1 it is sufficient to show that the corresponding

face angles of P and P′are equal, i.e., βj,k= β′

face angles by taking βj,k−β′

(i.e., Lemma 2.1) to the normal images T∗

(5) we obtain the following result.

{k1,k2}and (1) implies that the corresponding edges of the

{i,k}| i ∈ Ik} of the edges of Fkand F′

kare

α{i,k}

2

) and of equal length

j,k. So, let us compare those

j,k. Now, applying the Legendre-Cauchy lemma

vjand T∗

v′

jand using (3) as well as

Sublemma 2.3 Let vj,1 ≤ j ≤ v be an arbitrary vertex of the standard

ball-polyhedron P. Then either there are at least four sign changes in the

cyclic sequence of the face angle differences {βj,k− β′

the vertex vjof P or the cyclic sequence in question consists of zeros only.

j,k| vj∈ Fk} around

According to (1) (resp., (4)) Fk(resp., F′

of the unit sphere S(xk) (resp., S(x′

the spherical convex hull Fk(resp., F

{v′

property that Fk ⊂ Fk (resp., F

denotes the angular measure of the interior angle of Fk (resp., F

vertex vj(resp., v′

that the corresponding side lengths of Fk and F′

βj,k−β′

applied to Fkand F

k) is a spherically convex subset

k)) for any 1 ≤ k ≤ f and therefore

′

k) of the vertices {vj| j ∈ Jk} (resp.,

k) on S(xk) (resp., S(x′

′

k⊂ F′

j| j ∈ Jk}) of Fk(resp., F′

k) clearly possesses the

k). Moreover, if βj,k(resp., β

′

j,k)

′

k) at the

j), then (1) and (4) imply again in a straightforward way

kare equal furthermore,

j,k= βj,k−β

′

j,kholds for any vertex vj,j ∈ Jkof Fk. Thus, Lemma 2.1

′

kproves the following statement.

Sublemma 2.4 Let Fk,1 ≤ k ≤ f be an arbitrary face of the standard ball-

polyhedron P. Then either there are at least four sign changes in the cyclic

sequence of the face angle differences {βj,k−β′

Fkof P or the cyclic sequence in question consists of zeros only.

j,k| vj∈ Fk} around the face

Finally, let us take the medial graph G of P with “vertices” corresponding

to the edges of P and with “edges” connecting two “vertices” if the corre-

sponding two edges of P are adjacent (i.e., share a vertex in common) and lie

on the same face of P. So, if the “edge” of G “connects” the two edges of P

that lie on the face Fkof P and have the vertex vjin common enclosing the

face angle βj,k, then we label the “edge” in question of G by sign(βj,k−β′

j,k),

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Page 9

where sign(δ) is +,− or 0 depending on whether δ is positive, negative or

zero. Thus, using Sublemma 2.3 and Sublemma 2.4, one can apply the sign

counting lemma (i.e., Lemma 2.2) to the dual graph G∗of G concluding

in a straightforward way that βj,k− β′

Theorem 1.1.

j,k= 0. This finishes the proof of

3Underlying Truncated Delaunay Complex

of a Ball-Polyhedron

In this section we introduce some additional notations and tools that are

needed for our proof of Theorem 1.4.

First, recall that a convex polyhedron of E3is a bounded intersection of

finitely many closed half-spaces in E3. A polyhedral complex in E3is a finite

family of convex polyhedra such that any vertex, edge, and face of a member

of the family is again a member of the family, and the intersection of any

two members is empty or a vertex or an edge or a face of both members.

Second, let us give a detailed construction of the so-called truncated De-

launay complex of a ball-polyhedron, which is going to play an important role

in the proof of Theorem 1.4. We leave the proofs of the claims mentioned

here to the reader partly because they are straightforward and partly because

they are also well known (see for example, [2] or [9]).

The farthest-point Voronoi tiling corresponding to a finite set C := {c1,

...,cn} in E3is the family V := {V1,...,Vn} of closed convex polyhedral

sets Vi := {x ∈ E3: ?x − ci? ≥ ?x − cj?

1 ≤ i ≤ n. (Here a closed convex polyhedral set means a not necessarily

bounded intersection of finitely many closed half-spaces in E3.) We call the

elements of V farthest-point Voronoi cells. In the sequel we omit the words

“farthest-point” as we do not use the other (more popular) Voronoi tiling:

the one capturing closest points.

It is known that V is a tiling of E3. We call the vertices, (possibly un-

bounded) edges and (possibly unbounded) faces of the Voronoi cells of V

simply the vertices, edges and faces of V.

The truncated Voronoi tiling corresponding to C is the family Vtof the

closed convex sets {V1∩B[c1],...,Vn∩B[cn]}. Clearly, from the definition

it follows that Vt= {V1∩ P,...,Vn∩ P} where P := B[c1] ∩ ... ∩ B[cn].

We call the elements of Vttruncated Voronoi cells.

for all j ?= i,1 ≤ j ≤ n},

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Page 10

Next, we define the (farthest-point) Delaunay complex D assigned to the

finite set C = {c1, ...,cn} ⊂ E3. It is a polyhedral complex on the vertex set

C. For an index set I ⊂ {1,...,n}, the convex polyhedron conv{ci| i ∈ I}

is a member of D if and only if there is a point p in ∩i∈IViwhich is not

contained in any other Voronoi cell, where conv{·} stands for the convex hull

of the corresponding set. In other words, conv{ci| i ∈ I} ∈ D if and only if

there is a point p ∈ E3and a radius ρ > 0 such that {ci| i ∈ I} ⊂ bd(B(p,ρ))

and {ci| i / ∈ I} ⊂ B(p,ρ), where B(p,ρ) stands for the open ball having

radius ρ and center point p in E3. It is known that D is a polyhedral complex

moreover, it is a tiling of conv{c1, ...,cn} by convex polyhedra. The more

exact connection between the Voronoi tiling V and the Delaunay complex

D is described in the following statement. (In what follows, dim(·) refers to

the dimension of the given set, i.e., dim(·) stands for the dimension of the

smallest dimensional affine subspace containing the given set.)

Lemma 3.1 Let C = {c1, ...,cn} ⊂ E3be a finite set, and V = {V1,...,

Vn} be the corresponding Voronoi tiling of E3.

(V) For any vertex p of V there there exists an index set I ⊂ {1,...,n}

with dim({ci| i ∈ I}) = 3 such that conv{ci| i ∈ I} ∈ D and p =

∩i∈IVi. Vica versa, if I ⊂ {1,...,n} with dim({ci| i ∈ I}) = 3 and

conv{ci| i ∈ I} ∈ D, then ∩i∈IViis a vertex of V.

(E) For any edge E of V there exists an index set I ⊂ {1,...,n} with

dim({ci | i ∈ I}) = 2 such that conv{ci | i ∈ I} ∈ D and E =

∩i∈IVi. Vica versa, if I ⊂ {1,...,n} with dim({ci| i ∈ I}) = 2 and

conv{ci| i ∈ I} ∈ D, then ∩i∈IViis an edge of V.

(F) For any face F of V there exists an index set I ⊂ {1,...,n} of cardi-

nality 2 such that conv{ci| i ∈ I} ∈ D and F = ∩i∈IVi. Vica versa, if

I ⊆ {1,...,n} of cardinality 2 and conv{ci| i ∈ I} ∈ D, then ∩i∈IVi

is a face of V.

Finally, we define the truncated Delaunay complex Dtassigned to C

similarly to D. For an index set I ⊂ {1,...,n}, the convex polyhedron

conv{ci| i ∈ I} ∈ D is a member of Dtif and only if there is a point p in

∩i∈I(Vi∩ B[ci]) which is not contained in any other truncated Voronoi cell.

Recall that the truncated Voronoi cells are contained in the ball-polyhedron

P = B[c1] ∩ ... ∩ B[cn]. Thus, conv{ci| i ∈ I} ∈ Dtif and only if there

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exists a point p ∈ P and a radius ρ > 0 such that {ci| i ∈ I} ⊂ bd(B(p,ρ))

and {ci| i / ∈ I} ⊂ B(p,ρ).

4Proof of Theorem 1.4

Let P := ∩f

by the reduced family {B[xk] | 1 ≤ k ≤ f} of f ≥ 4 unit balls. Let CP:=

conv{x1,...,xf} be the center-polyhedron of P in E3. The following is a key

observation for the proof of Theorem 1.4 presented in this section.

k=1B[xk] be an arbitrary normal ball-polyhedron of E3generated

Lemma 4.1 Any normal ball-polyhedron P of E3is a standard ball-polyhed-

ron with its face lattice being dual to the face lattice of its center-polyhedron

CP.

Proof: First, let us take an arbitrary circumscribed sphere say, S(x,δ) of

the flower-polyhedron P∪= ∪f

By assumption 0 < δ < 2. Let I ⊂ {1,...,f} denote the set of the indices

of the unit balls {B[xk] | 1 ≤ k ≤ f} that are tangent to S(x,δ) (with the

remaining unit balls lying inside the circumscribed sphere S(x,δ)). Also,

let V and D (resp., Vtand Dt) denote the Voronoi tiling and the Delaunay

complex (resp., the truncated Voronoi tiling and the truncated Delaunay

complex) assigned to the finite set C := {x1, ...,xf}. It follows from the

definition of S(x,δ) in a straightforward way that dim({xi| i ∈ I}) = 3 and

conv{xi | i ∈ I} ∈ D. Thus, part (V ) of Lemma 3.1 clearly implies that

x = ∩i∈IViis a vertex of V. Furthermore, 0 < δ < 2 yields that x ∈ int(P)

and therefore x is a vertex of Vtas well and conv{xi| i ∈ I} ∈ Dt, where

int(·) stands for the interior of the corresponding set. Second, it is easy to

see via part (V ) of Lemma 3.1 that each vertex x of V is in fact, a center

of some circumscribed sphere of the flower-polyhedron P∪. Thus, we obtain

that the vertex sets of V and Vtare identical (lying in int(P)) and therefore

the polyhedral complexes D and Dtare the same, i.e., D ≡ Dt. Finally,

based on this and using Lemma 3.1 again, we get that the vertex-edge-face

structure of the normal ball-polyhedron P is dual to the face lattice of the

center-polyhedron CP= conv{x1,...,xf} (which is tiled by the polyhedral

complex D ≡ Dt). Thus, indeed P is a standard ball-polyhedron with the

desired face lattice, finishing the proof of Lemma 4.1.

k=1B[xk] having center point x and radius δ.

✷

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Page 12

For the rest of the discussions in this section we use the notations from

the proof of Theorem 1.1. Thus, P = ∩f

combinatorially equivalent normal ball-polyhedra with equal corresponding

face angles in E3and our goal is to show that P is congruent to P′.

Lemma 4.1 and the analogue of Alexandrov’s theorem for standard ball-

polyhedra (proved in [4]) imply that P and P′have equal corresponding

inner dihedral angles.Hence, for any k with 1 ≤ k ≤ f we have that

the corresponding face angles (resp., inner dihedral angles) of the faces Fk

and F′

all i ∈ Ik). So, the face Fk (resp., F′

convex subset of the unit sphere S(xk) (resp., S(x′

{E{i,k} | i ∈ Ik} (resp., {E′

circular arc of Euclidean radius sin

2

spherical length of E{i,k}(resp., E′

k=1B[xk] and P′= ∩f

k=1B[x′

k] are two

kare equal, i.e., βj,k = β′

j,kfor all j ∈ Jk (resp., α{i,k}= α′

k) of P (resp., P′) is a spherically

k)) with the family of edges

{i,k}| i ∈ Ik}), where E{i,k} (resp., E′

α{i,k}

. Let x{i,k}(resp., x′

{i,k}).

{i,k}for

{i,k}) is a

{i,k}) denote the

Lemma 4.2 Assume that P and P′are two combinatorially equivalent nor-

mal ball-polyhedra with equal corresponding face angles and inner dihedral

angles in E3. Let Fk,1 ≤ k ≤ f be an arbitrary face of the normal ball-

polyhedron P. Then either there are at least four sign changes in the cyclic

sequence of the corresponding edge length differences {x{i,k}−x′

around the face Fkof P or the cyclic sequence in question consists of zeros

only.

{i,k}| i ∈ Ik}

Proof: Using (1) and (4) as well as the duality part of Lemma 4.1 we get

that the points {zi,k| i ∈ Ik} ⊂ S(xk) (resp., {z′

in spherically convex position on S(xk) (resp., S(x′

point of {zi,k | i ∈ Ik} (resp., {z′

convex hull sconvS(xk){zi,k| i ∈ Ik} (resp., sconvS(x′

points {zi,k| i ∈ Ik} (resp., {z′

let ˆ zi,k (resp., ˆ z′

on S(xk) (resp., S(x′

ˆF′

xk,z−xk? ≤ 0 for all z ∈ Fk} (resp., (F′

0 for all z′∈ F′

As the proof of the following statement is rather straightforward we leave its

technical details to the reader.

i,k| i ∈ Ik} ⊂ S(x′

k)) meaning that every

k)) are

i,k| i ∈ Ik}) is a vertex of the spherical

k){z′

i,k| i ∈ Ik}) on S(xk) (resp., S(x′

i,k) be the diametrically opposite point to zi,k (resp., z′

k)). Moreover, letˆFk:= sconvS(xk){ˆ zi,k| i ∈ Ik} (resp.,

k){ˆ z′

k)∗:= {y′∈ S(x′

k}) be the polar set of Fk(resp., F′

i,k| i ∈ Ik}) of the

k)). Now,

i,k)

k:= sconvS(x′

i,k| i ∈ Ik}). Furthermore, let (Fk)∗:= {y ∈ S(xk) | ?y −

k) | ?y′−x′

k) on S(xk) (resp., S(x′

k,z′−x′

k? ≤

k)).

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Sublemma 4.3 Assume that P and P′are two combinatorially equivalent

normal ball-polyhedra with equal corresponding face angles and inner dihedral

angles in E3. Let Fk,1 ≤ k ≤ f be an arbitrary face of the normal ball-

polyhedron P. Then the following properties hold.

(a) (Fk)∗= sconvS(xk){Ck[ˆ zi,k,π

sconvS(x′

(Fk)∗⊂ S(xk) (resp., (F′

lar arcs described in more details as follows.

2−

] | i ∈ Ik}.

k)∗⊂ S(x′

α{i,k}

2

] | i ∈ Ik} and similarly, (F′

Moreover, the boundary of

k)) consists of geodesic and circu-

k)∗=

k){C′

k[ˆ z′

i,k,π

2−

α{i,k}

2

(b) Every Ck[ˆ zi,k,π

tributes to the boundary of (Fk)∗(resp., (F′

spherical length say, ˆ x{i,k}(resp., ˆ x′

sign(ˆ x{i,k}− ˆ x′

2−

α{i,k}

2

] (resp., C′

k[ˆ z′

i,k,π

2−

α{i,k}

2

]) with i ∈ Ik con-

k)∗) along a circular arc of

{i,k}) satisfying sign(x{i,k}−x′

{i,k}) =

{i,k}).

(c) For every vertex vj ∈ Fk (resp., v′

geodesic arc of spherical length π−βj,kon the boundary of (Fk)∗(resp.,

(F′

Ck[ˆ zi,k,π

2

] and Ck[ˆ zi0,k,π

C′

2

]), where E{i0,k},E{i,k}(resp., E′

the two edges of Fk(resp., F′

j∈ F′

k) with j ∈ Jk there exists a

k)∗) that is tangent at its end points to the closed spherical caps

2−

k[ˆ z′

k) meeting at the vertex vj(resp., v′

α{i,k}

2−

α{i0,k}

2

] (resp., C′

k[ˆ z′

{i0,k},E′

i,k,π

2−

{i,k}) denote

α{i,k}

2

] and

i0,k,π

2−

α{i0,k}

j).

Now, we claim that the spherically convex polygonsˆFk⊂ S(xk) andˆF′

S(x′

holds because, part (c) of Sublemma 4.3 implies that the geodesic distance

between the center points of the closed spherical caps Ck[ˆ zi,k,π

Ck[ˆ zi0,k,π

2

] on the unit sphere S(xk) is determined by the spherical

length (namely, π−βj,k) of their common non-separating tangential geodesic

arc. Finally, let ˆ γi,k(resp., ˆ γ′

angle ofˆFk(resp.,ˆF′

of Sublemma 4.3 imply that sign(x{i,k}−x′

all i ∈ Ik. Thus, by applying the Legendre-Cauchy lemma (i.e., Lemma 2.1)

to the spherically convex polygonsˆFk andˆF′

finished.

Finally, let us take the edge graph G of the normal ball-polyhedron P

and label the edge corresponding to E{i,k}by sign(x{i,k}− x′

Lemma 4.2 and the sign counting lemma (i.e., Lemma 2.2) applied to the

dual graph of G complete our proof of Theorem 1.4.

k⊂

k) have equal corresponding spherical side lengths. Indeed, this claim

2−

α{i,k}

2

] and

2−

α{i0,k}

i,k) denote the angular measure of the interior

k) at the vertex ˆ zi,k(resp., ˆ z′

{i,k}) = −sign(ˆ γi,k− ˆ γ′

i,k). Then parts (b) and (c)

i,k) holds for

k, our proof of Lemma 4.2 is

✷

{i,k}). Clearly,

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Page 14

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K´ aroly Bezdek

Department of Mathematics and Statistics, University of Calgary, Canada,

Department of Mathematics, University of Pannonia, Veszpr´ em, Hungary,

Institute of Mathematics, E¨ otv¨ os University, Budapest, Hungary,

E-mail: bezdek@math.ucalgary.ca

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