The Steiner ratio conjecture of Gilbert and Pollak is true.

Critical points for the Steiner ratio

Let T be a minimal spanning tree for the points, i.e. T is a shortest tree with vertices precisely at xl. X2, ... , Xn. (A Steiner tree is allowed to have additional vertices). IfLs, LT denote the lengths of S, T respectively, then the Steiner ratio p = LsILT. The Steiner ratio conjecture was introduced by Gilbert and Pollak: [2] and asked if p::::: "';3/2 for all sets of points. We were able t...

An Approach for Proving Lower Bounds: Solution of Gilbert-Pollak's Conjecture on Steiner Ratio

'Let {gi(z)}iEr be a family of finitely many continuous functions on a polytope X. Consider the problem of minimizing the function f(z) = maxiGrgi(2) on X. Many problems about lower bounds can be reduced to this general form. In this paper, we show that if every gi(z) is a concave function, then the minimum value of f(z) is achieved at finitely many special points in X. As an application, we so...

On the Steiner Ratio in 3-Space

| The \Steiner minimal tree" (SMT) of a point set P is the shortest network of \wires" which will suuce to \electrically" interconnect P. The \minimum spanning tree" (MST) is the shortest such network when only intersite line segments are permitted. The \Steiner ratio" (P) of a point set P is the length of its SMT divided by the length of its MST. It is of interest to understand which point set...

Minimax and Its Applications: Revisit the Proof of Gilbert-pollak Conjecture

The Steiner ratio for the dual normed plane

A minimum Steiner tree for a given set X of points is a network interconnecting the points of X having minimal possible total length. The Steiner ratio for a metric space is the largest lower bound for the ratio of lengths between a minimum Steiner tree and a minimum spanning tree on the same set of points in the metric space. Du et al. (1993) conjectured that the Steiner ratio on a normed plan...