On Steiner Minimal Trees with Rectilinear Distance

Abstract

We consider Steiner minimal trees in the plane with rectilinear distance. The rectilinear distance $d(p_1 ,p_2 )$ between two points $p_1 $, $p_2 $ is $| {x_1 - x_2 } | + | {y_1 - y_2 } |$, where the $(x_i ,y_i )$ are the Cartesian coordinates of the $p_i $. For a given finite set P of points, let $l_s $ denote the length of a Steiner minimal tree and $l_m $ the length of a minimal spanning tree. The main result of the memorandum is that ${l_s / l_m } \geqq \frac{2}{3}$.