A variational approach to the Steiner network problem

Abstract

Suppose n points are given in the plane. Their coordinates form a 2 n-vector X. To study the question of finding the shortest Steiner network S connecting these points, we allow X to vary over a configuration space. In particular, the Steiner ratio conjecture is well suited to this approach and short proofs of the cases n=4, 5 are discussed. The variational approach was used by us to solve other cases of the ratio conjecture (n=6, see [11] and for arbitrary n points lying on a circle). Recently, Du and Hwang have given a beautiful complete solution of the ratio conjecture, also using a configuration space approach but with convexity as the major idea. We have also solved Graham's problem to ..