DOI: 10.1007/s00454-006-1262-y

Discrete Comput Geom 36:503–509 (2006) Discrete & Computational

Geometry© 2006 Springer Science+Business Media, Inc.

On Distinct Distances from a Vertex of a Convex Polygon

Adrian Dumitrescu

Computer Science, University of Wisconsin–Milwaukee,3200 N. Cramer Street, Milwaukee, WI 53211, USAad@cs.uwm.edu

Dedicated to Janos Pach on the occasion of his 50th birthday

Abstract. Given a set P of n points in convex position in the plane, we prove thatthere exists a point p ∈ P such that the number of distinct distances from p is at least�(13n − 6)/36�. The best previous bound, �n/3�, from 1952, is due to Moser.

1. Introduction

A set of points in the plane is in general position if no three points are collinear. A finiteset of points is in convex position if the points are the vertices of a convex polygon.

The following famous problem was raised by Erdos [3] in 1946. What is the minimumnumber of distinct distances determined by n points in the plane? Denoting this numberby g(n), he conjectured that g(n) = �(n/√log n), and showed that this bound is attainedby the

√n×√n integer grid. The best known lower bound,≈ �(n0.8641), is the last in a

sequence of recent results due to Solymosi and C. Toth [13], Tardos [14], and Katz andTardos [9]. See also the related work of Pach and Tardos [12].

If the n points are in convex position, then the minimum number of distinct dis-tances determined is much larger. The following classical result of Moser [10] is wellknown:

Theorem 1 (Moser). Let P be a set of n points in convex position in the plane. Thenthere exists a point p ∈ P such that the number of distinct distances from p is atleast �n/3�.

Following in part the presentation in Pach and Agarwal’s book [11, page 206], letgconv(n) denote the minimum number of distinct distances determined by a set of n points

504 A. Dumitrescu

in convex position. Altman [1], [2] proved that

gconv(n) =⌊n

2

⌋.

The bound is tight, and attained for a regular n-gon.Erdos conjectured that the following stronger statement is also true. Every convex

n-gon has a vertex such that the number of distinct distances from this vertex is at leastn/2. If true, this would be again best possible, as a regular n-gon shows. We make astep in this direction by improving the bound in Theorem 1, a result which is by nowmore than 50 years old. Somewhat surprisingly, most of the ingredients present in ourproof were known for many years.

Theorem 2. Let P be a set of n points in convex position in the plane. Then there existsa point p ∈ P such that the number of distinct distances from p is at least �(13n−6)/36�.

As mentioned in [11], it seems likely that the above result of Altman can be generalizedto any set of points in general position. That is, if ggen(n) denotes the minimum numberof distinct distances determined by a set of n points in general position in the plane, itcan be conjectured that

ggen(n) = gconv(n) =⌊n

2

⌋.

The best result in this direction is due to Szemeredi [4] (see also page 207 of [11]):

Theorem 3 (Szemeredi). Let P be a set of n points in general position in the plane.Then there exists a point p ∈ P such that the number of distinct distances from p is atleast �(n − 1)/3�.

Note that the above bound for points in general position is essentially the same as thatfor points in convex position (Theorem 1), from which it can differ by at most one. OurTheorem 2 introduces some distinction between the results for points in general positionand the presumably easier case of points in convex position.

A result related to Theorem 1 is the following. Let h(n) be the largest integer h suchthat every convex polygon with n vertices has a vertex p so that the next h verticesclockwise from p, or the next h vertices counterclockwise from p, are successivelyfarther from p. Erdos and Fishburn [5] have proved that h(n) = n/3 + 1 for n ≥ 4.Their proof of the lower bound slightly extends Moser’s approach which was showingthat h(n) ≥ �n/3� (and thereby improves Moser’s lower bound by one in the case whenn is a multiple of three).

2. Proof of Theorem 2

Let C be a circular disk, and let pq be a chord of C . The chord pq divides the disk intotwo parts, the smaller of which is called a cap; if pq is a diameter of C , then both partsare considered caps. Lemma 1 below is used by Moser in proving his �n/3� lower bound(see page 206 of [11]).

On Distinct Distances from a Vertex of a Convex Polygon 505

Lemma 1 [10]. Let P be a set of m points in convex position inside a closed cap ofa disk C determined by a chord pq , such that p, q ∈ P . Then all the m − 1 distancesbetween p and the other points of P are distinct.

The first part in our proof is the same as in Moser’s proof (see pages 206–207 of[11]). Let C be the smallest disc containing all points in P . If only two points p, q ∈ Plie on the boundary of C , then pq must be a diameter, and at least one of the two closedcaps determined by pq contains at least �n/2� + 1 points of P . By Lemma 1, there areat least �n/2� distinct distances from p, and Theorem 2 follows in this case.

If C has more than two points on its boundary, then one can choose three of them,p, q, r ∈ P , so that no angle of�pqr is obtuse (for otherwise C would not be minimum;the detailed argument appears in [10]). Since P is in convex position, no element of Pis in the interior of �pqr . Thus, all of P is contained in the three circular caps C1, C2,and C3 determined by the chords pq, qr , and r p, respectively. Denote by mi the numberof points in cap Ci , i = 1, 2, 3. Then

m1 + m2 + m3 = n + 3.

We now adapt Szemeredi’s proof to the situation when the points are in convex positionand are included in the three caps as outlined above. The original proof (conforming withthe presentation on pages 207–208 of [11]) goes as follows.

Assume that for every point p ∈ P , the number of distinct distances from p is atmost k. This means that every point q = p lies on one of at most k concentric circlesC1(p), . . . ,Ck(p) centered at p. Let I be the number of isosceles triangles determinedby P , where an equilateral triangle is counted with multiplicity three. Then,

I =∑p∈P

k∑i=1

(|Ci (p) ∩ P|2

), (1)

which attains its minimum if the points of P\{p} are distributed among the circles Ci (p)as uniformly as possible. In particular,

I ≥ nkn − 1

k

(n − 1

k− 1

)/2. (2)

On the other hand, every segment qr can be the base of at most two isosceles trianglesdetermined by P , for otherwise the perpendicular bisector of qr would pass through atleast three points of P . Thus,

I ≤ 2

(n

2

). (3)

Putting these two inequalities together, one gets k ≥ �(n − 1)/3�.We now return to the proof of Theorem 2, and first improve the upper bound on I .

Two key facts in our proof are the following easy generalization of Lemma 1 and itscorollary:

506 A. Dumitrescu

p1 pm

d

fxg

e

zy

Fig. 1. Points in a circular cap; proof of Lemma 2.

Lemma 2. Let P = {p1 · · · pm} be a set of m points in convex position inside a closedcap of a disk C determined by chord p1 pm , where the points are labeled in clockwiseorder. Then for any i ∈ {1, . . . ,m}, (a) all the m− i distances between pi and the pointspj ( j > i) of P (which follow it in clockwise order) are distinct; (b) all the i−1 distancesbetween pi and the points pj ( j < i) of P (which precede it in clockwise order) aredistinct.

Proof. For simplicity assume that p1 pm is horizontal and the circular cap lies above it.We have to show that each isosceles triangle xyz, with xy = xz, is of the form pi , pj , pk ,where 1 ≤ i < j < k ≤ m and pi pj = pj pk ; in other words, x lies between y andz. Assume otherwise, say y and z follow x in clockwise order (the case when y and zprecede x in clockwise order is similar).

Put α = x yz. Denote by g and e the points of intersection of the line xy with thecircle. Denote by d and f the points of intersection of the line yz with the circle. SeeFig. 1. Clearly, α < 90◦, as one of the equal angles of an isosceles triangle. On the otherhand, as an angle formed by two intersecting chords

α = α1 + α2

2,

where α1, α2 are the measures of the circular clockwise arcs de and f g. Clearly, α1 ≥ 0◦,and by the convexity of P , α2 ≥ 180◦ (i.e., p1 is on or below the line ge, and pm is onor below the line d f ). This gives α ≥ 90◦, which is a contradiction.

We say that p determines an isosceles triangle�pqr if pq = pr ; thus, an equilateraltriangle is determined by all its three vertices. We also say that �pqr is determined byP if p, q, r ∈ P .

Corollary 1. Let P be a set of m points in convex position inside a closed cap of adisk C determined by a chord pq , such that p, q ∈ P . Then the number I of isoscelestriangles determined by P is at most (m − 1)2/4.

On Distinct Distances from a Vertex of a Convex Polygon 507

Proof. Let j ≤ (m+1)/2. We claim that pj determines at most j−1 isosceles triangles.Indeed, by Lemma 2, there are no isosceles triangles�pj pk1 pk2 determined by pj , wherej < k1, k2. Similarly, there are no isosceles triangles�pi1 pi2 pj determined by pj , wherei1, i2 < j . Assume that �pi pj pk1 and �pi pj pk2 are two isosceles triangles determinedby pj , where i < j < k1 and i < j < k2. Then �pj pk1 pk2 , where j < k1, k2, is also anisosceles triangles determined by pj , contradicting the above observation. This provesour claim.

Similarly, if j ≥ (m+1)/2, pj determines at most m− j isosceles triangles. Therefore,the total number of isosceles triangles determined by P is bounded as follows:

For even m,

I ≤ 2∑

2≤ j≤m/2

( j − 1) = (m − 2)m

4≤ (m − 1)2

4.

For odd m,

I ≤ 2∑

2≤ j≤(m−1)/2

( j − 1)+ m − 1

2= (m − 1)2

4.

We continue with the proof of Theorem 2, and denote by P a set of n points in convexposition. By Corollary 1, for each i ∈ {1, 2, 3}, at least

(mi

2

) − (mi − 1)2/4 segmentswhose endpoints are in Ci are the base of at most one isosceles triangle determined byP . Consequently, the number I of isosceles triangles determined by P is

I ≤ 2

(n

2

)−

3∑i=1

{(mi

2

)− (mi − 1)2

4

}≤ 2

(n

2

)−

3∑i=1

m2i − 1

4. (4)

By the Cauchy–Schwarz inequality,

3∑i=1

m2i − 1

4≥ 1

4

(3

(n + 3

3

)2

− 3

)= n2 + 6n

12.

Plugging this bound into (4), we get that

I ≤ 2

(n

2

)− n2 + 6n

12= 11n2 − 18n

12. (5)

Comparing (2) and (5), one gets k � 6n/17. We now obtain a better lower bound than(2), which in turn gives a better result. For a given p ∈ P , denote by I (p) the number ofisosceles triangles determined by p. We recall that I (p) attains its minimum when thepoints of P\{p} are distributed on the (at most k concentric) circles C1(p), . . . ,Ck(p)centered at p as evenly as possible. Since we can assume that �n/3� ≤ k ≤ n/2, thismeans that each circle contains either two or three points. Let a (resp. b) be the numberof circles centered at p that pass through exactly two (resp. three) points of P\{p}. Wehave a + b ≤ k and 2a + 3b = n − 1. Solving for a and b yields

a ≤ 3k − (n − 1),

b ≥ (n − 1)− 2k.

508 A. Dumitrescu

Then the number of isosceles triangles I (p) determined by p is

I (p) ≥ a + 3b = (2a + 3b)− a ≥ (n − 1)− (3k − (n − 1)) = 2(n − 1)− 3k,

and, consequently,

I =∑p∈P

I (p) ≥ n(2n − 2− 3k). (6)

Comparing (5) and (6) yields

k ≥⌈

13n − 6

36

⌉,

as promised.

3. Concluding Remarks

Let t (p) be the number of distinct distances from vertex p to the other vertices of aconvex n-gon P . The t-sequence of P , viewed cyclically, is (t (p1), t (p2), . . . , t (pn))

with p1, p2, . . . , pn in clockwise (or counterclockwise) order. Let T (P) = t (p1) +t (p2)+ · · · + t (pn), and

Tn = min{T (P): Pis a convex n-gon}.

We conclude by recalling two related conjectures on convex n-gons, C2 and C3, posedby Erdos and Fishburn [6]–[8], vis-a-vis conjecture C1, the one discussed in this paper:

C1. (Erdos) Some vertex has at least n/2 distinct distances to other vertices.C2. (Fishburn) Each of at least n/2 vertices has at least n/2 distinct distances to

other distances.C3. (Erdos and Fishburn) Tn =

(n2

)for all n ≥ 3.

Clearly C2 is a strengthening of C1. Since C3 implies that the minimum average t (p)is (n − 1)/2, C3 is a strengthening of C1 as well.

References

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5. P. Erdos and P. Fishburn, A postscript on distances in convex n-gons, Discrete & Computational Geometry,11 (1994), 111–117.

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8. P. Fishburn, Distances in convex polygons, in: The Mathematics of Paul Erdos, vol. 2 (R. Graham andJ. Nesetril, editors), Springer-Verlag, New York, 1996, pages 284–293.

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Received June 30, 2005. Online publication September 29, 2006.