Reducing the Steiner Problem in a Normed Space

Abstract

Consider a set P of points in a normed space whose unit sphere is a d-dimensional symmetric polytope with 2d extreme points. This paper proves that there always exists a Steiner minimum tree whose Steiner points are located only at points whose coordinates appear in points of P. This generalizes a recent result of Snyder on d-dimensional rectilinear space, which itself extends Hanan’s well-known and much quoted result on the rectilinear plane. Furthermore, the proof in this paper is much simpler than Snyder’s proof, even considerably shorter than Hanan’s proof. A consequence of this result is that the Steiner problem for P in such a space is reduced to a Steiner problem on graphs and is solvable by any existing Steiner graph algorithms. The paper also conjectures that such a reduction is impossible if the polytope has more than $2d$ extreme points and provides partial support for the conjecture.