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: In this paper we introduce ball-polyhedra as finite intersections of congruent balls in Euclidean 3-space. We define their duals and study their face-lattices. Our main result is an analogue of Cauchy’s rigidity theorem.European Journal of Combinatorics 01/2006; · 0.66 Impact Factor - 01/1991;
- Around the proof of the Legendre-Cauchy lemma on convex polygons, Siberian Math. I Kh, Sabitov . 740-762.
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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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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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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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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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Page 11
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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Page 13
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,
13
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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