The Steiner Ratio of $\mathcal{L}_p^3$

Abstract

We consider Steiner’s Problem in $\mathcal{L}_p^3$ , which is a three-dimensional space equipped with p-norm. Steiner’s Problem is the “Problem of shortest connectivity”, that means, given a finite set N of points in the plane, search for a network interconnecting these points with minimal length. This shortest network must be a tree and is called a Steiner Minimal Tree (SMT). It may contain vertices different from the points which axe to be connected. Such points are called Steiner points. If we do not allow Steiner points, that means, we only connect certain pairs of the given points, we get a tree which is called a Minimum Spanning Tree (MST) for N. Steiner’s Problem is very hard as well in combinatorial as in computational sense, but on the other hand, the determination of an MST is simple. Consequently, we are interested in the greatest lower bound for the ratio between the lengths of these both trees:

$$ m(3,p): = inf\left\{ {\frac{{L(SMT for N)}} {{L(MST for N)}}:N \subseteq {\text{ }}\mathcal{L}_p^3 is a finite set} \right\} $$

, which is called the Steiner ratio (of $\mathcal{L}_p^3$ ,).

We look for estimates for m(3,p), depending on the parameter p, and determine general upper bounds for the Steiner ratio of $\mathcal{L}_p^3$ ,.