In this paper we present two results dealing with the number of (• k)-facets of a set of points:

•² In R2, we use the notion of ²-net to give structural properties of sets that achieve the optimal lower bound 3¡k+22 ¢ of (• k)-edges for a ¯xed 0 • k • bn=3c ¡ 1;²

•In Rd, we show that for k < bn=(d + 1)c the number of (• k)-facets of a set of n points in general position is at least (d + 1)¡k+dd ¢, and that this bound is tight in that range

Aichholzer, Oswin

Garcia Lopez de Lacalle, Jesus

Orden Martín, David

Ramos, Pedro

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

More about lower bounds for the number of
(≤ k)-facets ?
O. Aichholzer1??, J. Garc´ıa2???, D. Orden3†, and P. Ramos3‡
1 Institute for Software Technology. Graz University of Technology. Graz, Austria.
oaich@ist.tugraz.at
2 Escuela Universitaria de Informa´tica. Universidad Polite´cnica de Madrid. Madrid,
Spain. jglopez@eui.upm.es
3 Departamento de Matema´ticas. Universidad de Alcala´. Alcala´ de Henares, Spain.
{david.orden,pedro.ramos}@uah.es
Abstract. In this paper we present two results dealing with the number of (≤ k)-
facets of a set of points:
• In R2, we use the notion of ²-net to give structural properties of sets that achieve
the optimal lower bound 3
(
k+2
2
)
of (≤ k)-edges for a fixed 0 ≤ k ≤ bn/3c − 1;
• In Rd, we show that for k < bn/(d+ 1)c the number of (≤ k)-facets of a set of n
points in general position is at least (d+ 1)
(
k+d
d
)
, and that this bound is tight in
that range.
Key words: (≤ k)-edges, (≤ k)-facets, ²-nets
1 Introduction
In this paper we deal with the problem of giving lower bounds to the number
of (≤ k)-facets of a set of points S: An oriented simplex with vertices at points
of S is said to be a k-facet of S if it has exactly k points in the positive side
of its affine hull. Similarly, the simplex is said to be an (≤ k)-facet if it has at
most k points in the positive side of its affine hull. If S ⊂ R2, a k-facet of S
is usually named a k-edge.
The number of k-facets of S is denoted by ek(S), and Ek(S) =
∑k
j=0 ej(S)
is the number of (≤ k)-facets (the set S will be omitted when it is clear
? Research partially supported by grant HU2007-0017.
?? Research partially supported by the FWF (Austrian Fonds zur Fo¨rderung der Wis-
senschaftlichen Forschung) under grant S09205, NFN Industrial Geometry.
??? Research partially supported by grant MCYT TIC2002-01541.
† Research partially supported by grants MTM2005-08618-C02-02 and S-0505/DPI/0235-
02.
‡ Research partially supported by grants TIC2003-08933-C02-01 and S-0505/DPI/0235-02.
2 O. Aichholzer, J. Garc´ıa, D. Orden, and P. Ramos
from the context). Giving bounds on these quantities, and on the number of
the companion concept of k-sets, is one of the central problems in Discrete
and Computational Geometry, and has a long history that we will not try to
summarize here. Chapter 8.3 in [4] is a complete and up to date survey of
results and open problems in the area.
Regarding lower bounds for Ek(S), which is the main topic of this paper,
the problem was first studied by Edelsbrunner et al. [6] due to its connections
with the complexity of higher order Voronoi diagrams. In that paper it was
stated that, in R2,
Ek(S) ≥ 3
(
k + 2
2
)
(1)
and it was given an example showing tightness for 0 ≤ k ≤ bn/3c−1. The proof
used circular sequences but, unfortunately, contained an unpluggable gap, as
pointed out by Lova´sz et al. [8]. A correct proof, also using circular sequences,
was independently found by A´brego and Ferna´ndez-Merchant [1] and Lova´sz
et al. [8]. In both papers a strong connection was discovered between the
number of (≤ k)-edges and the number of convex quadrilaterals in a point set
S. The interested reader can go through the extensive online bibliography by
Vrt’o [9] where the focus is on the problem of crossing numbers of graphs.
The lower bound in Equation 1 was slightly improved for k ≥ bn3 c by
Balogh and Salazar [3], again using circular sequences. Using different tech-
niques, and based on the observation that it suffices to proof the bound for
sets with triangular convex hull, we have recently shown [2] that, in R2,
Ek(S) ≥ 3
(
k + 2
2
)
+
k∑
j=bn
3
c
(3j − n+ 3). (2)
If n is divisible by 3, this expression can be written as
Ek(S) ≥ 3
(
k + 2
2
)
+
(
k − n3 + 2
2
)
.
In this paper we deal with two different problems related to lower bounds
for Ek: In Section 2, we study structural properties of those sets in R2 that
achieve the lower bound in Equation 1 for a fixed 0 ≤ k ≤ bn/3c − 1. We use
the notion of ²-net to prove that if Ek(S) is minimum for a given k, then Ej(S)
is also minimum for every 0 ≤ j < k. In Section 3 we study the d-dimensional
version of the problem and show that, for a set of n points in general position
in Rd,
Ek(S) ≥ (d+ 1)
(
k + d
d
)
, for 0 ≤ k < b nd+1c, (3)
and that this bound is tight in that range. To the best of our knowledge, this
is the first result of this kind in Rd.
More about lower bounds for the number of (≤ k)-facets 3
2 Optimal sets for (≤ k)-edge vectors
Given S ⊂ R2, let us denote by Ek(S) the set of all (≤ k)-edges of S, hence
Ek(S) is the cardinality of Ek(S). Throughout this section we consider k ≤
bn3 c − 1. Recall that for a fixed such k, Ek(S) is optimal if Ek(S) = 3
(
k+2
2
)
.
Recall also that, by definition, a j-edge has exactly j points of S in the positive
side of its affine hull, which in this case is the open half plane to the right of
its supporting line.
We start by giving a new, simple, and self-contained proof of the bound
in Equation 1, using a new technique which will be useful in the rest of the
section. The following notions are presented in Rd for further use in Section 3.
Definition 1 ([7]). Let S be a set of n points and H a family of sets in Rd.
A subset N ⊂ S is called an ²-net of S (with respect to H) if for every H ∈ H
such that |H ∩ S| > ²n we have that H ∩N 6= ∅.
Definition 2. A simplicial ²-net of S ⊂ Rd is a set of d + 1 vertices of the
convex hull of S that are an ²-net of S with respect to closed half-spaces. A
simplicial 12 -net will be called a simplicial half-net.
Lemma 1. Every set S ⊂ R2 of n points has a simplicial half-net.
Proof. Let T be a triangle spanned by three vertices of the convex hull of S.
An edge e of T is called good if the closed half plane of its supporting line
which contains the third vertex of T , contains at least n2 points from S. T is
called good if it consists of three good edges. Clearly, the vertices of a good
triangle are a simplicial half-net of S.
Let T be an arbitrary triangle spanned by vertices of the convex hull of S
and assume that T is not good. Then observe that only one edge e of T is
not good and let v be the vertex of T not incident to e. Choose a point v′ of
the convex hull of S opposite to v with respect to e. Then e and v′ induce a
triangle T ′ in which e is a good edge. If T ′ is a good triangle we are done.
Otherwise we iterate this process. As the cardinalities of the subsets of vertices
of S considered are strictly decreasing (the subsets being restricted by the half
plane induced by e), the process terminates with a good triangle. uunionsq
Theorem 1. For every set S of n points and 0 ≤ k < bn−22 c we have Ek(S) ≥
3
(
k+2
2
)
.
Proof. The proof goes by induction on n. From Lemma 1, we can guarantee
the existence of T = {a, b, c} ⊂ S, an 12 -net made up with vertices of the
convex hull. Let S′ = S \ T and consider an edge e ∈ Ek−2(S′). We observe
that T cannot be to the right of e: there are at least n2 points on the closed
half-plane to the left of e and that would contradict the definition of 12 -net.
Therefore, e ∈ Ek(S). If we denote by ET k(S) the set of (≤ k)-edges of S
adjacent to points in T , we have that
4 O. Aichholzer, J. Garc´ıa, D. Orden, and P. Ramos
Ek−2(S′) ∪ ET k(S) ⊂ Ek(S). (4)
There are 2(k+1) (≤ k)-edges incident to each of the convex hull vertices
a, b, c (which can be obtained rotating a ray based on that vertex). We observe
that at most three edges of ET k(S) might be incident to two points of T (those
of the triangle T ) and that the union in Equation 4 is disjoint. Therefore, using
the induction hypothesis we have
Ek(S) ≥ Ek−2(S′) + 3 + 6k ≥ 3
(
k
2
)
+ 3 + 6k = 3
(
k + 2
2
)
. (5)
uunionsq
Corollary 1. Let S be a set of n points, T = {a, b, c} a simplicial half-net
of S and S′ = S \ T . If Ek(S) = 3
(
k+2
2
)
then:
(a)Ek−2(S′) = 3
(
k
2
)
.
(b)A k-edge of S is either a (k− 2)-edge of S′ or is adjacent to a point in T .
Proof. If Ek(S) = 3
(
k+2
2
)
, both inequalities in Equation 5 are tight. Therefore
Ek−2(S′) = 3
(
k
2
)
and Equation 4 becomes Ek−2(S′)∪ET k(S) = Ek(S) (disjoint
union) which trivially implies part (b). uunionsq
Theorem 2. If Ek(S) = 3
(
k+2
2
)
, then S has a triangular convex hull.
Proof. We prove the statement by induction over k. For k = 0 nothing has
to be proven, so let k = 1, assume that E1 = 9 and let h = |CH(S)|. We
have h 0-edges and at least h 1-edges (two per convex hull vertex, but each
edge might be counted twice). Thus E1 = 9 ≥ 2h and therefore h ≤ 4. Assume
now h = 4. Then at most two 1-edges can be counted twice, namely the two
diagonals of the convex hull. Thus we have 4+8− 2 = 10 (≤ 1)-edges and we
conclude that if E1 = 9, then S has a triangular convex hull.
For the general case consider k ≥ 2, let T = {a, b, c} be the simplicial half-
net guaranteed by Lemma 1 and let S′ = S \ T . From Corollary 1, part (a),
we know that Ek−2(S′) = 3
(
k
2
)
and, by induction, we may assume that S′ has
a triangular convex hull. Moreover, from part (b), no (k − 1)-edge of S′ can
be an (≤ k)-edge of S and, therefore, any (k − 1)-edge of S′ must have two
vertices of T on its positive side. Consider the six (k− 1)-edges of S′ incident
to the three convex hull vertices of S′: See Figure 1, where the supporting
lines of these (k − 1)-edges are drawn as dashed lines and S′ is depicted as
the central triangle. Each cell outside S′ in the arrangement of the supporting
lines contains a number counting the (k−1)-edges considered which have that
cell on their positive side. A simple counting argument shows that the only
way of placing the three vertices a, b, c of T such that each (k − 1)-edge of S′
drawn has three of them on its positive side is to place one in each cell labeled
with a 4. We conclude that no vertex of S′ can be on the convex hull of S and
the theorem follows. uunionsq
More about lower bounds for the number of (≤ k)-facets 5
S
′
a b
c
4
4
4
3
3
3
3
3
3
2
2
2
Fig. 1. Each (k − 1)-edge of S′ incident to a convex hull vertex of S′ (supporting
lines are shown as dashed lines) has two vertices of T on its positive side.
Corollary 2. If Ek(S) = 3
(
k+2
2
)
, then the outermost dk2e layers of S are
triangles.
Proof. From the optimality of Ek(S) and using the same argument as in the
proof of Theorem 2, it follows that we can iteratively remove the outermost
dk2e layers to obtain optimal subsets, which, by Theorem 2, have triangular
convex hulls. uunionsq
Theorem 3. If Ek(S) = 3
(
k+2
2
)
, then Ej(S) = 3
(
j+2
2
)
for every 0 ≤ j ≤ k.
Proof. We prove the theorem by induction on k. For k = 0, 1 the theorem is
equivalent to Theorem 2, so let k ≥ 2. It is sufficient to show that optimality
of Ek(S) implies optimality of Ek−1(S), as the theorem follows by induction.
Let T be the vertices of CH(S) (which is a triangle as guaranteed by
Theorem 2) and let S′ = S \ T . As in Theorem 1 we have
Ek−3(S′) ∪ ET k−1(S) ⊂ Ek−1(S).
Observe that Ek−2(S′) is optimal, as guaranteed by Corollary 1 and this
implies optimality of Ek−3(S′) by induction. |ET k−1(S)| is also optimal be-
cause the convex hull of S is the triangle T . Therefore, to prove optimality of
Ek−1(S) it only remains to show that no (k−2)-edge of S′ can be a (k−1)-edge
of S.
So let e be a (k−2)-edge of S′ and let p and q be the vertices of the convex
hull of S′ incident to e or on its positive side. The existence of p and q is
guaranteed by Corollary 1, part (b). Without loss of generality, assume that
the edge pq is horizontal with the remaining vertices of S′ above it, see Figure 2
for the rest of the proof. Let `1 be the (k − 1)-edge of S′ incident to p which
has q on its positive side and `2 the (k−1)-edge incident to q and having p on
its positive side. The boundary chain is the lower envelope of `1, pq, and `2.
6 O. Aichholzer, J. Garc´ıa, D. Orden, and P. Ramos
We claim that e does not intersect the boundary chain and lies above it. If e
is incident to p or q then the claim is obviously true. Otherwise observe that e
has to intersect the supporting lines of both considered (k − 1)-edges in the
interior of S′, as otherwise there would be too many vertices on the positive
side of e. But then again e lies above the boundary chain and the claim follows.
S
′
a b
p q
e ∈ Ek−2(S
′)
`2 ∈ Ek−1(S
′)
`1 ∈ Ek−1(S
′)
Fig. 2. All (k− 2)-edges of S′ (supporting lines are shown as dotted lines) lie above
the (bold) lower envelope.
From the proof of Theorem 2 we know that two of the vertices of the convex
hull of S have to lie below our boundary chain (below the (k − 1)-edges, see
a and b in Figure 2) and thus on the positive side of e. Therefore e has at
least k vertices of S on its positive side and does not belong to Ek−1(S). We
conclude that Ek−1(S) is optimal and the theorem follows. uunionsq
Corollary 3. Let 0 ≤ k ≤ bn3 c − 1. If Ek(S) = 3
(
k+2
2
)
, then ej(S) = 3(j + 1)
for 0 ≤ j ≤ k.
3 A lower bound for (≤ k)-facets in Rd
Throughout this section, S ⊂ Rd will be a set of n points in general position.
We remind that ek(S) and Ek(S) denote, respectively, the number of k-facets
and the number of (≤ k)-facets of S. The main result of this section is a lower
bound for the number of (≤ k)-facets of a set of n points in general position
in Rd in the range 0 ≤ k < b nd+1c.
The proof follows the approach in Theorem 1, using the fact that every set
of points has a centerpoint: a point c ∈ Rd is a centerpoint of S if no open
halfspace that avoids c contains more than d dnd+1e points of S (see [5]).
Theorem 4. Let S be a set of n ≥ d + 1 points in Rd in general position.
Then the following bound holds tightly in the given range of k
Ek(S) ≥ (d+ 1)
(
k + d
d
)
if 0 ≤ k < b nd+1c.
More about lower bounds for the number of (≤ k)-facets 7
Proof. The proof uses induction on n and d. The base case for n = d + 1 is
obvious and for d = 2 is just Equation 1.
Let k < b nd+1c and let c be a centerpoint of S. Let us consider a simplex T
with vertices in the convex hull of S and containing c and let S′ = S \T . From
the definition of centerpoint, it follows that no open halfspace that avoids T
contains more than d dnd+1e − 1 points or, equivalently, every closed halfspace
containing T has at least b nd+1c + 1 points. We denote by Ejk(S) the set of
(≤ k)-facets of S adjacent to exactly j vertices of T , and Ejk(S) will be the
cardinality of Ejk(S).
For j = 0, we observe that E0k−d(S′) ⊂ E0k (S), because a closed halfspace
containing at most k points cannot contain all the vertices of T . Because
k − d ≤ bn−(d+1)d+1 c − 1, we can apply induction on n and get
E0k(S) ≥ E0k−d(S′) ≥ (d+ 1)
(
(k − d) + d
d
)
= (d+ 1)
(
k
d
)
.
For 1 ≤ j ≤ d, let Tj be a subset of j vertices of T and let Spi be the
projection from Tj of S \T onto the (d−j)-dimensional subspace pi defined by
the points in T \ Tj : a point p ∈ S \ T is mapped to the intersection between
the j-flat defined by p and Tj and the (d− j)-flat defined by points in T \ Tj .
Using the general position assumption, it is easy to see that the intersection
has dimension zero. If the intersection were empty, we could slightly perturb p
without changing the number of (≤ k)-facets of S.
Now, if σ ⊂ Spi is an (≤ (k−d+j))-facet of Spi, then σ∪Tj is an (≤ k)-facet
of S (as before, a halfspace containing at most k points of S cannot contain
all the vertices of T ). Because k − d + j ≤ b nd+1c − 1 ≤ b n−jd−j+1c − 1, we can
apply induction on d and n, obtaining that there are at least
(d− j + 1)
(
k − d+ j + (d− j)
d− j
)
= (d− j + 1)
(
k
d− j
)
(≤ k)-facets of S adjacent to Tj . Summing on all the subsets of j points of T ,
we get
Ejk(S) ≥
(
d+ 1
j
)
(d− j + 1)
(
k
d− j
)
,
and, finally,
Ek(S) ≥
d∑
j=0
(
d+ 1
j
)
(d− j + 1)
(
k
d− j
)
= (d+ 1)
(
k + d
d
)
.
As for tightness, the example showing that the bound 3
(
k+2
2
)
is tight for
0 ≤ k ≤ bn3 c − 1 in the planar case [6] can be extended to Rd: Consider
d + 1 rays in Rd emanating from the origin and with the property that any
hyperplane containing one of them leaves on each open halfspace at least one
8 O. Aichholzer, J. Garc´ıa, D. Orden, and P. Ramos
of the remaining rays. For instance, we could take the rays defined by the
origin and the vertices of a regular simplex inscribed in the unit d-sphere.
Let n = (d + 1)m and put chains C1, . . . , Cd+1 with m points on each ray,
slightly perturbed to achieve general position. uunionsq
4 Open problems
Our bound in Rd generalizes that in Equation 1 except for the restriction
k < bn/(d + 1)c, which stems from the underlying technique, namely using
the centerpoint of a set, and can probably be removed. An alternative proof
of Theorem 4, using a simplicial half-net instead of a centerpoint, would be
sufficient to extend the bound to the whole range of k. Therefore, it is a chal-
lenging task to extend Lemma 1 to dimension d, as the following conjecture
states:
Conjecture 1. Every point set S ⊂ Rd has a simplicial half-net.
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crossing number. Graphs and Combinatorics, 21:3 (2005), 293–300.
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number of (≤ k)-edges and the rectilinear crossing number of Kn. Discrete and
Computational Geometry 38:1 (2007), 1–14.
[3] J. Balogh and G. Salazar. Improved bounds for the number of (≤ k)-sets, con-
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[5] H. Edelsbrunner. Algorithms in Combinatorial Geometry, Springer Verlag, 1987.
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