Optimal Steiner Points

Abstract

Given three points P, Q, and R in the plane, the shortest path connecting the points using only straight lines between the points is equal to the sum of the shorter two sides of APQR. By introducing a fourth point interior to APQR when all its angles are less than 120?, a shorter network is obtained. There is a point S in APQR such that the sum of the Euclidean distances of S from the three vertices is minimized. This point is called the Steiner point of the triangle, after the mid-19th-century Swiss geometer Jacob Steiner, an early investigator of the question of minimal networks among a finite number of points [2]. For example, in an equilateral triangle of side 1, a minimal path that connects the vertices and stays on the perimeter has length 2, whereas a network of vertices and Steiner point S has length 3 = 1.732, still more than half the perimeter. By introducing a well-known Minkowski metric and constructing the Steiner point with respect to that metric, the length of the minimal network can be improved for any three vertices to half the length of the perimeter. Such a Steiner point is optimal in the sense that the lengths in the network could not be shortened and still satisfy the triangle inequality required in a metric space. A point S in the metric space (d, X) is said to be an optimal Steiner point of A PQR if the d-length of the network connecting P, Q, R, and S is half the d-length of the perimeter of A PQR. We shall see that, if the metric is Minkowski, all triangles have optimal Steiner points if and only if the defining unit ball is a parallelogram. We start by noting some familiar facts. Given any three positive real numbers [a, b,c], some or all of which may be the same, the triple can be the distances between three points in the Euclidean plane, denoted by E2, as long as the sum of any two is greater than or equal to the third. [1, 1, 1] can be realized in E2 as the distances between the vertices P1, P2, P3 of any equilateral triangle of side length 1. [3,4,5] and [1, 1, 2] will do, but [1, 1, 2.001] will not. These are particular examples of the more general concept of a metric set. A set F of n abstract points together with a distance function d: F X F -> [0, oo) satisfying the axioms for a metric space is called a metric set of n points. Any metric set of three points can be embedded in the plane with the Euclidean distance function as a triangle or as collinear points. When a fourth point is introduced, there are six distances. Denoting d(Pi, Pj) by dij for i, j = 1,2,3,4, we always keep the order [dI2, d23, d3l, dI4, d24, d34]I Going back to an equilateral triangle of side 1 and vertices P1, P2, and P3 in E2, let us look for a fourth point P4 that has distance 1 from each vertex. The metric set {P1, P2, P3, P4} will have associated with it the list of six distances [1, 1, 1, 1, 1, 1]. There is no such point in the plane. If we are willing to go to Euclidean 3-space, the four points can be realized as vertices of a tetrahedron with all sides length 1.