The Number of Furthest Neighbour Pairs of a Finite Planar Set

Abstract

Geometric problems involving a finite set of points in the plane have intrigued mathematicians for many years. As far back as 1893, J. J. Sylvester [4] posed the following problem: If S is a set of points in the plane, not all collinear, must there exist a line that contains exactly two points of S? This problem remained unsolved for forty years until it was rediscovered by P. Erdos in 1933 and solved a few days later by T. Gallai. This problem and a myriad of other famous problems in plane geometry are discussed in the fascinating paper by V. Klee [2]. In this note, we are concerned with results about maximum distances between pairs of points in the plane. An early problem of this type was posed by Hopf and Pannwitz [1] in 1934: What is the maximum number of pairs of points that can realize the diameter, the maximum distance between points of a set? The answer to this question appears in [3] and is the same as the number of points in the set. This is achieved by placing one point at the center of a circle of unit radius and the remaining points on a part of the boundary of the circle so that the two most distant points also are at distance 1 apart. We shall state and solve a variation of this problem. Consider a set N of n points in the Eucidean plane indexed by the integers 1, 2,.. ., n. A point j is called a furthest neighbour of a point i if d(i, j) = maxl,k n d(i,k), where d is the Euclidean distance function. Each point has at least one, and possibly several, furthest neighbours. Let n, denote the number of furthest neighbours of point i. Finally let