A Helly-Type Theorem for Unions of Convex Sets
Discrete Comput Geom
A Helly-Type Theorem for Unions of Convex Sets¤
J. Matousˇek 0
0 Department of Applied Mathematics, Charles University , Malostranske ́ na ́m. 25, 118 00 Praha 1 , Czech Republic
We prove that for any d, k ¸ 1 there are numbers q D q.d; k/ and h D h.d; k/ such that the following holds: Let K be a family of subsets of the d-dimensional Euclidean space, such that the intersection of any subfamily of K consisting of at most q sets can be expressed as a union of at most k convex sets. Then the Helly number of K is at most h. We also obtain topological generalizations of some cases of this result. The main result was independently obtained by Alon and Kalai, by a different method. ¤ This research was supported by Czech Republic Grant GACˇ R 201/94/2167, Charles University Grants Nos. 351 and 361 and by EC Cooperative Action IC-1000 (project ALTEC: Algorithms for Future Technologies). A preliminary version appeared in Proceedings of the 10th Annual ACM Symposium on Computational Geometry, 1995.
Let Rd denote the d -dimensional Euclidean space. A famous theorem of Helly,
discovered in 1913, asserts that if F is a finite family of convex sets in Rd such that any d C 1
or fewer sets of F have a point in common, then also the intersection of all sets of F is
nonempty. Over the years, a vast body of analogs and generalizations of this result has
been accumulated in the literature, see [Ec] for a recent survey.
A general scheme of a Helly-type theorem is captured by the following definition.
Let K be an arbitrary family of sets. We say that K has Helly number h (h is a natural
number) if the following holds for any finite subfamily F µ K: if the intersection of
any subfamily of F consisting of ·h sets is nonempty, then T F (the intersection of all
sets of F ) is nonempty. Thus, the family C of all convex sets in Rd has Helly number
d C 1.
We prove the following Helly-type result:
Theorem 1. For any k ¸ 1, d ¸ 1 there are numbers q D q.d; k/ and h D h.d; k/
with the following property. Let K be a family of subsets of Rd such that the intersection
of any subfamily of K consisting of q or fewer sets can be expressed as a union (not
necessarily disjoint) of at most k convex sets. Then K has Helly number at most h.
This result was independently obtained by Alon and Kalai [AKa]. Their method is
more complicated than the one presented here (and quite different), on the other hand, they
obtain the result as a consequence of a powerful theorem concerning piercing numbers
of certain families, which they prove by a method developed by Alon and Kleitman
The estimates of the numerical values of the Helly numbers and of the numbers
q.d; k/ following from our proof of Theorem 1 are quite large. For instance, for unions
of convex sets in the plane, the current proof gives estimates h.2; 2/ · 20, h.2; 3/ · 90,
h.2; 4/ · 231, etc. Small improvements are possible by refining the argument, but we
are still probably quite far from the best values. It would be interesting to get better upper
bounds and/or some nontrivial lower-bound examples.
We also prove a partial topological analog of Theorem 1. Before stating it, we recall
the notion of a j -connected topological space. A topological space X is said to be
.¡1/-connected if it is nonempty. For j D 0; 1; 2; : : : ; X is j -connected if it is . j ¡
1/connected and moreover any continuous mapping f of the sphere S j into X can be
extended to a continuous mapping fN: B jC1 ! X , where B jC1 is the ball bounded
by S j . In particular, a 0-connected set means just path-connected and nonempty, and
1connected means 0-connected and simply connected. Intuitively, a 1-connected subspace
of R3 may not have any “tunnels” but may have “bubbles.”
Theorem 2. For any d ¸ 2, k ¸ 1 there is a number h D h.d; k/ (where h.d; 1/ ·
d C 2 for d even and h.d; 1/ · d C 3 for d odd) with the following property. Let K be
a family of subsets of Rd such that the intersection of any nonempty finite subfamily of
K has at most k path-connected components, each of which is .dd=2e ¡ 1/-connected.
(In particular, for d D 2 we require that any intersection of a finite subfamily of K has
at most k path-connected components.) Then K has Helly number at most h.
The author learned of a proof of the simple special case d D 2, k D 1 of Theorem 2
(in a slightly different context) from Nina Amenta. We suspect it was also observed by
others a long time ago, but currently we have no explicit reference.
Currently we do not know whether it is sufficient to restrict the assumption to at most
q-wise intersections for some bounded q D q.d; k/ (as can be done in Theorem 1). We
believe that there might be a general result saying that if all at most q-wise intersections
of sets of a family K in Rd have “topological complexity” bounded by k, then the Helly
number of K is bounded in terms of d and k; here q would depend on k; d. A definition
of “topological complexity” of a set suitable in this context is yet to be found; perhaps
it could be the minimum number of simplices needed to triangulate the set, or some
number derived from the homology groups.
The rest of the paper is structured as follows. In Section 2 we describe some previous
work related to our result and we mention a recent new motivation for studying
Hellytype problems. This part is not needed for understanding the proofs. The proofs are given
in Sections 3 (Theorem 1) and 4 (Theorem 2). Section 5 presents two examples whose
meaning is discussed in Section 2 below.
Motivation and Related Work
LP-Type Problems. Recently, a new motivation for studying Helly-type results came
from geometric optimization algorithms. Sharir and Welzl [SW] defined a class of
optimization problems, the so-called LP-type problems, which encompasses linear
programming, convex programming, and other natural geometric optimization problems.
They gave an efficient algorithm for solving problems in this class (provided that certain
primitive operations can be implemented efficiently for the problem in question). Also
several other algorithms can be applied for LP-type problems; see [G], [C], [MSW], and
[Ma]. A crucial parameter in these problems is their dimension, and this is closely related
to the Helly number of suitable set systems in Rd . Roughly speaking, if it is desired to
show that the above-mentioned algorithms work fast for an optimization problem with
some set of constraints, then it is necessary to bound the Helly number of certain derived
set systems by a small number. These relations have been investigated by Amenta [A1],
[A2]. We believe that studying Helly-type properties is important for understanding the
structure of LP-type problems and potentially also for developing and analyzing yet
more-efficient algorithms for these problems, or perhaps proving lower bounds for such
Previous Work: Disjoint Unions. Gru¨nbaum and Motzkin [GM] considered Helly-type
theorems for disjoint unions of convex sets. They conjectured the following:
Theorem 3. Let K be a family of sets in Rd such that the intersection of any its subfamily
of size at most k can be expressed as a disjoint union of k closed convex sets. Then the
Helly number of K is at most k.d C 1/.
In this theorem it is important to assume that not only the members of K, but also their
·k-wise intersections can be expressed as disjoint unions of at most k closed convex sets.
For instance, the family of all unions of disjoint pairs of closed convex sets has no finite
Helly number. For the same reason, in Theorem 1 we need an assumption concerning
·q-wise intersections of sets of K (rather than sets of K only).
Examples show that the bound k.d C 1/ for the Helly number in Theorem 3 is the
best possible in general. Gru¨nbaum and Motzkin [GM] proved Theorem 3 for k D 2, the
k D 3 case was established by Larman [L], and the general case was proved by Morris
[Mo].1 A short elegant proof of Theorem 3 was given by Amenta [A1].2
1 To which Eckhoff [Ec] remarks “However, Morris’ proof : : : is extremely involved and the validity of
some of his arguments is, at best, doubtful.”
2 Gru¨nbaum and Motzkin in fact conjectured a more general result in an abstract setting. Let B be a family
of sets which is intersectional (that is, B1 \ B2 2 B for any B1; B2 2 B) and nonadditive (that is, no finite
disjoint union of at least two nonempty sets of B belongs to B) and has Helly number at most h. Let [B]k
denote the family of all disjoint unions of at most k members of B. The conjecture of [GM] can be formulated
We note that our Theorem 1 implies that, in particular, under the assumptions of
Theorem 3, the Helly number is bounded, but the proof only provides a much worse
bound than the correct value k.d C 1/. On the other hand, in our Theorem 1 the unions
need not be disjoint and the convex sets considered need not be closed.
The closedness assumption in Theorem 3 is important for the value of the Helly
number. We consider a family K in the plane such that each set as well as the intersection
of each two sets is a disjoint union of two convex sets (not necessarily closed). Gru¨nbaum
and Motzkin [GM] construct such a family with Helly number 9 (instead of 6 as would
be obtained with convex sets replaced by closed convex sets). On the other hand, it is
still possible that the Helly number of any such K is bounded; Gru¨nbaum and Motzkin
conjecture that 9 is the correct value. Theorem 1 implies a weaker statement, namely
that if it is assumed that the intersection of any at most three sets of K is a union of at
most two convex sets, then the Helly number is bounded by a constant.
Depth of Intersections and n-Convexity. As was observed in [GM], if K is a family
such that the intersection of any ·k sets of K is a union of at most k closed convex sets,
then the intersection of any number of sets of K is a union of at most k closed convex sets.
This leads to the question whether something similar holds in the situation of Theorem 1.
Specifically, we might ask for which d and k there are numbers c D c.d; k/, K D K .d; k/
such that the following statement holds:
Statement 4. Suppose that K is a family in Rd such that the intersection of any at most
c sets of K can be expressed as a union of at most k convex sets. Then the intersection
of any finite subfamily of K is a union of at most K convex sets.
A simple example, presented as Example 9 in Section 5 below, shows that, with K D k
(which would be the strongest form), Statement 4 holds for no c, even in the simplest
case d D k D 2. Example 10 then implies that, for d ¸ 4, k ¸ 2, Statement 4 holds with
no c, K at all.
Example 9 was noted by Pavel Valtr. Later the author found out that a related question
has been investigated in several previous papers (see [PS], [BK], and references therein)
under the heading of n-convexity, and the examples were essentially known. In particular,
the four-dimensional Example 10 uses almost the same construction as an example due to
Perles (published in [KPS]). We thus include the examples for the reader’s convenience
For d D 2 and perhaps for d D 3, Statement 4 might be true with large enough c
and K . For d D 2, the results of Perles and Shelah [PS] easily imply that Statement 4
µk C 1¶
holds, after replacing “convex sets” by “closed convex sets,” with c.2; k/ · ,
as follows: If K is a family of sets such that the intersection of any at most k sets of K belongs to [B]k , then the
Helly number of K is at most kh. The family of all closed (resp. open) convex sets in Rd is both intersectional
and nonadditive. On the other hand, nonadditivity fails for the family of all convex sets. The k D 2; 3 cases
were proved in the above-mentioned papers [GM] and [L], respectively; for larger k, only Morris’ proof seems
to be available: Amenta’s proof of Theorem 3 can also be stated in an abstract setting, but different from the
one just described.
K .2; k/ · k6. A forthcoming paper [MV] improves the bound k6 to O.k3/, and shows
that, for d D 2, Statement 4 holds for all k (with suitable c.2; k/ and K .2; k/). The case
d D 3 seems to be wide open.
Topological Helly-Type Theorems. Helly’s theorem on convex sets has various
topological generalizations. Helly himself gave a topological version of his theorem in [H].
A modern proof and some generalizations were given by Debrunner [D]. To state some
of them, we recall a few more topological notions.
A topological space X is called a homology cell if it is nonempty and its (singular,
reduced) homology groups of all dimensions vanish; in particular, convex sets are
homology cells. More generally, X is called j -acyclic if all its homology groups up to
dimension j vanish. We remark that a j -connected topological space is j -acyclic, and
for j ¸ 2 a j -cyclic 1-connected space is j -connected (but acyclic spaces exist which
are not 1-connected).
Helly’s result can be rephrased as follows (see [D]): Let F be a finite family of open
subsets of Rd such that the intersection of any at most d members of F is a homology
cell, and the intersection of any d C 1 members is nonempty. Then T F is a homology
cell; in particular, it is nonempty. As a consequence, assuming that the intersection of
any at most d members of a family K of open sets in Rd is a homology cell, we get that
the Helly number of K is d C 1. Debrunner [D] shows that it is enough to assume that
the j -wise intersections of sets of F are .d ¡ j /-acyclic, j D 1; 2; : : : ; d C 1. He also
generalizes the results to families of open subsets of an arbitrary d-manifold.
In this context, Theorem 2 is of some interest also for k D 1, since the topological
theorems just mentioned require that the sets in the considered family are homology cells,
while our result allows them to have “holes,” more precisely, a nonzero homology in
dimensions dd=2e through d ¡1. On the other hand, we assume .dd=2e¡1/-connectedness
for all intersections, which is a very strong requirement.
3. Proof for Unions of Convex Sets
In this section we prove Theorem 1.
Let [n] denote the set f1; 2; : : : ; ng. For a set X and a natural number t , let
denote the set of all t -element subsets of X (we sometimes call them t -sets).
We need a suitable d-dimensional generalization of the well-known fact that a planar
graph with n vertices has only O.n/ edges. We set b D dd=2e C 1.
Lemma 5. Let d and ® > 0 be fixed, and let n D n.d; ®/ be sufficiently large. Let P
µ P¶ µn¶
be an n-point set in Rd , and let S µ
Then disjoint sets S; S0 2 S exist such that conv.S/ \ conv.S0/ D ;.
be a family of b-sets of P with jSj ¸ ®
We remark that, for our proof, it would be sufficient to have this lemma, e.g., with
d C 1 instead of b, only we get somewhat worse values for q.d; k/ and h.d; k/. Such a
statement is explicitly proved by Alon et al. [ABFK] (see also [BFL]).
For d D 3, Lemma 5 is a special case of the results of Dey and Edelsbrunner [DE].
Dey and Pach [DP] consider extensions of a similar method into higher dimensions; they
explicitly prove the statement of the Lemma with d instead of b, and it seems that their
method can be extended to yield the lemma itself.
Another proof for a general dimension can be given along the lines of Alon et al.
[ABFK]; we only sketch the method here. Consider the hypergraph H with vertex set
V .H/ D f.i; j /I i D 1; 2; 3; j D 1; 2; : : : ; bg and edge set
E .H/ D ff.i1; 1/; .i2; 2/; : : : ; .ib; b/gI i1; i2; : : : ; ib 2 f1; 2; 3gg
(a complete b-partite b-uniform hypergraph). By a result of van Kampen [vK] (mentioned
in [S]), the simplicial complex whose maximal simplices are the edges of H cannot be
embedded into Rd so that no two vertex-disjoint simplices intersect. By the Erdo¨s–Stone
theorem, the hypergraph with vertex set P and edge set S contains a copy of H, see
[ABFK], and the lemma follows.
Our basic approach is to prove Theorem 1 by induction on the number of sets in the
considered finite family F . One problem here is that we only assume that the intersection
of any subfamily of size ·q is “nice,” i.e., is a union of k convex sets. As was mentioned
in Section 2 (see Statement 4 and also Example 10 in Section 5), this does not necessarily
mean that intersections of larger subfamilies must be “nice” in this sense. We circumvent
this by introducing another notion of “nice,” which behaves more regularly in this respect.
Definition 6. Let t; j be natural numbers, t ¸ j , and let X be a set in Rd . We say that
X has property P.t; j / if for any t -set T µ X a j -set J µ T exists such that the convex
hull of J is contained in X .
We note that if a set X is a union of at most k convex sets and t ¸ k. j ¡ 1/ C 1, then X
has property P.t; j / (by the pigeonhole principle). Property P.t; j / is more convenient
to work with for our purposes than the property “being a union of ·k convex sets,”
because of the following easy lemma:
property P.t; j /.
Lemma 7. Let K be a family of sets in Rd such that the intersection of any at most
µ t ¶
sets of K has property P.t; j /. Then the intersection of any subfamily of K has
hulls of its j -sets is fully contained in X . For each J 2
conv. J / 6µ FJ . Then property P.t; j / also fails for the intersection
Proof. Let X be the intersection of a subfamily F µ K, and suppose that property
P.t; j / fails for X , that is, a t -set T µ X can be found such that none of the convex
, fix a set FJ 2 F with
We fix t D k.b ¡ 1/ C 1, q D q.d; k/ D
with this value of q implies that the intersection of any at most q sets of sets of K has
property P.t; b/, and by Lemma 7 any intersection of sets of K has property P.t; b/.
Hence Theorem 1 is proved by establishing the following:
. Then the assumption of Theorem 1
4. Proof of the Topological Result
The proof of Theorem 2 begins similarly to the proof of Proposition 8. We consider a
family F D fF1; : : : ; Fng, with n D h C 1 sufficiently large, and points p1; : : : ; pn,
with pi belonging to all sets of F but possibly Fi . We set t D k C 1, and consider
Proposition 8. For any d ¸ 1, t ¸ b D dd=2e C 1 there is a number h D h.d; t / with
the following property. Let K be a family of subsets of Rd such that the intersection of
any of its finite subfamily has property P.t; b/. Then K has Helly number at most h.
Proof. Throughout the proof, d and t are treated as constants (in the O, Ä notation).
Let h be sufficiently large so that all estimates below are valid (by inspecting the current
proof and a proof of Lemma 5, a specific value for h D h.d; t / can be found).
To prove that K has Helly number h it is enough to show that if F µ K is any
subfamily of h C 1 sets of K such that the intersection of each h sets of F is nonempty,
then T F 6D ; (the case of a larger F is then handled by induction on the number of
elements of F ). Let the sets of F be numbered F1; F2; : : : ; Fn, n D h C 1. For each i ,
choose a point pi 2 T
Consider a t -set J 2
. All the points pj , j 2 J , belong to the intersection
X J D Tj2[n]nJ Fj . Since X J has property P.t; b/, we can fix a b-set K D K . J / µ J
such that 1K :D convf pj I j 2 K g is contained in X J (if there are more choices for
K . J / fix one arbitrarily). In this situation we assign the set J nK to the b-set K D K . J /
as a label.
As we fix the K . J / for each t -set J , labels are assigned to b-sets. One b-set K can
receive one or several labels, or no label at all. If L is one of the labels of K , then we
know that the .b ¡ 1/-simplex 1K is contained in all sets Fj with j 62 L [ K . We call
a b-set K good if it has at least one label but the intersection of all its labels is empty,
and call K bad otherwise. Thus, for a good b-set K , 1K is contained in all sets Fj with
j 62 K .
Each bad b-set is assigned at most
labels, and there are
thus at most O.nt¡1/ labels are assigned to bad b-sets. Since
altogether, at least Ä.nt / labels are assigned to good b-sets. A good b-set is assigned
O.nt¡b/ labels, therefore there are Ä.nb/ good b-sets.
By Lemma 5, there are two disjoint good b-sets K , K 0 with 1K \ 1K 0 6D ;. By the
above-mentioned property of a good b-set, 1K is contained in all sets Fj with j 62 K ,
and similarly for 1K 0 . Thus, ; 6D 1K \ 1K 0 µ Tn
jD1 Fj .
labels are assigned
µn ¡ b ¡ 1¶
t ¡ b ¡ 1
. We know that all points pj , j 2 J , belong to the intersection
X J D Ti2[n]nJ Fi , and this is a union of at most k path-connected components. Therefore,
there is a pair P. J / D f j1; j2g ½ J of indices such that pj1 and pj2 belong to a common
path-connected component of X J (in fact, there may be many pairs, so we fix one
arbitrarily for each t -set J ).
The t -set considered is linearly ordered, and the pair P. J / can be uniquely encoded
by specifying a pair of elements of the set f1; 2; : : : ; t g (we call it the type of the pair
P. J /). The number of possible types is
, the important fact is that it is bounded by
a function of t . In this way, each J 2
types, which can be viewed as coloring all t -sets on [n]. If m is a prescribed parameter
and n D n.m; t / is chosen sufficiently large, by Ramsey’s theorem (see, e.g., [GRS]) an
is assigned one of a bounded number of
m-element subset M µ [n] exists such that all t -sets J 2
our case this means that they all have the same type of the pair P. J /.
have the same color. In
The Planar Case. First we finish the proof for the case d D 2, which is somewhat
simpler than the case of an arbitrary dimension. Choose a fixed nonplanar graph G, say
G D K5, and let V be its vertex set and E its edge set. We assume that a set M µ [n]
has been selected as above, so that all t -sets have the same type of P. J /, and jM j is
sufficiently large (as we will see, jM j ¸ 5 C 10.t ¡ 2/ suffices). We illustrate the method
for k D 2 (then t D 3) and assuming that the type of the pair is ² ² ± , the other cases
are quite similar. To each vertex v 2 V we assign a point '.v/ 2 M and to each edge
e 2 E we assign a t -set 8.e/ of points of M , in such a way that:
(C1) For any two disjoint e; e0 2 E , we have 8.e/ \ 8.e0/ D ;, and if e 6D e0 share
a vertex v, then 8.e/ \ 8.e0/ D f'.v/g.
(C2) For each e D fu; vg 2 E , the points '.u/ and '.v/ are just the points of the
Such an assignment, for the particular type of the pair P. J / of 3-sets, may be chosen
as indicated in Fig. 1 (the solid circles represent the points '.v1/; : : : ; '.v5/, the open
circles the other points of M , and the triples are marked by the segments at various levels
connected to their points). For larger t and/or other types of the pairs P. J /, we proceed
similarly: First we map the vertices of G by ' to distinct points of M , leaving large
enough gaps among them. For each edge e D fu; vg, its 8-image consists of '.u/ and
'.v/ and t ¡ 2 other points of M , chosen so that they do not occur in the image of any
other edge, and so that P.8.e// D f'.u/; '.v/g.
Now by (C2), for each edge e D fu; vg 2 E , the points p'.u/ and p'.v/ lie in the same
path-connected component of the intersection X8.e/ D Tj2[n]n8.e/ Fj , so we can choose
a path 1e µ X8.e/ connecting p'.u/ and p'.v/. These paths together yield a drawing of
K5, so some two paths belonging to vertex-disjoint edges e; e0 cross. The intersection
of these paths belongs to all sets Fi with i 62 8.e/, and also to all Fi with i 62 8.e0/.
However, 8.e/ \ 8.e0/ D ; by (C1) and we are done.
mapping 8: E !
Arbitrary Dimension. To complete the proof of Theorem 2 for a general dimension d,
we need a finite dd=2e-dimensional simplicial complex S, which will play the role of the
nonplanar graph K5 used in the planar case. Precisely, we use the following property:
Whenever f : kSk ! Rd is a continuous mapping, vertex-disjoint simplices s; s0 2 S
exist such that f .ksk/ \ f .ks0k/ 6D ; (here kSk denotes the polyhedron of S, which is
the topological space of some geometric realization of S, and, for a simplex s 2 S, ksk
means the closed subset of kSk corresponding to the simplex s; see, e.g., [Mu] for an
introduction to simplicial complexes). We might use, e.g., the complex mentioned in the
proof sketch for Lemma 5, or we may apply a more well-known result of van Kampen
[vK] and Flores [F], which says that if S is the j -skeleton of the .2 j C 2/-dimensional
simplex, then it has the required property for d · 2 j .
Let a suitable simplicial complex S be fixed. We let G be its 1-skeleton, consisting
of the vertices and 1-simplices (edges) of S. Hence G is formally a one-dimensional
simplicial complex, but we may also regard it as a graph with vertex set V and edge set
E consisting of all one-dimensional simplices of S. We perform for G the construction
we did for K5 in the planar case. Thus, we have a sufficiently large set M µ [n] of
indices such that all t -sets of M have the same type, an injective mapping ': V ! M , a
satisfying (C1) and (C2), and a “drawing” of G in Rd , that is,
a continuous mapping f : kGk ! Rd . This f satisfies f .v/ D p'.v/ for all v 2 V and
f .kek/ µ X8.e/ for each edge e 2 E .
We extend the mapping f continuously to the whole kSk. First we extend the definition
of the mapping 8 to all simplices s 2 S. 8.s/ is already defined if s 2 S is an edge
(1-simplex). If s D fvg 2 S is a vertex, we simply put 8.s/ D f'.v/g, and for s 2 S of
dimension ¸ 2 we let
It is easy to check that this extended 8 satisfies:
(C10) For any two vertex-disjoint simplices s; s0 2 S, we have 8.s/ \ 8.s0/ D ;.
The continuous extension of the mapping f is constructed inductively. Suppose that f
has already been defined at all points of each ksk, where s 2 S is a simplex of dimension
less than j (for some j; 2 · j · dd=2e/, and moreover we have, for any such s,
f .ksk/ µ X8.s/:
We define f on j -dimensional simplices. Consider a j -dimensional simplex s 2 S.
Let k@sk denote the portion of ksk corresponding to proper faces of s. This set is
homeomorphic to the . j ¡ 1/-sphere S j¡1, and f .k@sk/ is contained in X8.s/, as follows
) applied on the . j ¡ 1/-faces of s. Since k@sk is path-connected, its image
is contained in a single path-connected component of X8.s/. By our assumptions, this
component is . j ¡ 1/-connected, so we can extend f continuously to the relative interior
of ksk, in such a way that the image is still contained in X8.s/. This finishes the induction
Having defined f on the whole kSk, by the choice of S we know there are two
vertexdisjoint faces s; s0 2 S with f .ksk/ \ f .ks0k/ 6D ;. By (
), we have f .ksk/ \ f .ks0k/ µ
X8.s/ \ X8.s0/ D X8.s/\8.s0/ D X; D T[n] Fi , by (C10). This concludes the proof.
In this section we present examples related to Statement 4 discussed in Section 2.
Example 9. For any odd integer n, sets F1; : : : ; Fn µ R2 exist such that the intersection
of any at most n ¡ 1 of them is a union of two convex sets, while Tn
iD1 Fi cannot be
expressed as a union of fewer than three convex sets.
Proof. First, let C be a regular convex n-gon and let v1; : : : ; vn denote its vertices
numbered along the circumference. We let Fi be C minus the relative interior of the edge
vi viC1 .vnC1 meaning v1). If the intersection Tn
iD1 Fi were a union of two convex sets,
there would be two consecutive vertices vi , viC1 belonging to the same convex set, but
this is impossible, since the interior of the edge vi viC1 is missing from the intersection.
On the other hand, for any intersection X of fewer than n of the Fi , there is one edge,
say vnv1, which is contained in X . Let A1 be the set X minus all vertices vi with i odd,
and let A2 be X minus all vertices vi with i even. Then we have X D A1 [ A2 and it is
easily checked that A1 and A2 are convex.
The example can be easily modified so that sets Fi are closed (or open). Consider the
midpoint mi of the edge vi viC1. Move mi a little bit toward the center of C , obtaining a
point mN i , and let Fi be C minus the interior of the triangle vi mN i viC1. Figure 2 illustrates
the construction for n D 5: the left part shows the set F1, and the right part the set
iD1 Fi .
Example 10. For any given integers c; K , there are n and sets F1; : : : ; Fn µ R4 such
that the intersection of any at most c of them is a union of two convex sets, while Tn
cannot be expressed as a union of fewer than K convex sets.
Proof. Let G be a K -chromatic graph such that any subgraph of G with at most c edges
is 2-colorable (the existence of such a graph follows from [Er], say). Let C be a cyclic
polytope in R4 with jV .G/j vertices, and suppose that its vertices are identified with the
vertex set V of G. Any two vertices of C are connected by an edge (one-dimensional
face) of C . Let e1; : : : ; en be the edges of G. For ei D fu; vg, we let Fi be C minus the
relative interior of the edge uv of the polytope C . The set Tn
iD1 Fi cannot be expressed
as a union of fewer than K convex sets, since this would induce a proper coloring of G
by less than K colors. On the other hand, if X D Tei 2E0 Fi , where E 0 is a set of at most
c edges of G, we let Â : V ! f1; 2g be a 2-coloring of the graph with edge set E 0. For
j D 1; 2 we set Aj D X nfv 2 V I Â .v/ 6D j g. As in the previous example, it may be
checked that the Aj are convex and cover X . A modification with all Fi closed or all Fi
open is again possible.
I would like to acknowledge two significant contributions to this paper by other people.
Nina Amenta brought the problems to my attention and she gave an argument which can
be interpreted as a proof of Theorem 2 for d D 2, k D 1; this inspired the basic approach
of all the proofs. Pavel Valtr pointed out Example 9, which directed me to using property
P .t ; j /. I also thank two anonymous referees for helpful suggestions concerning the
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