A new bound for the Steiner ratio

Abstract:

Let $V$ denote a given set of $n$ points in the euclidean plane. A Steiner minimal tree for $V$ is the shortest network (clearly, it has to be a tree) interconnecting $V$. Junctions of the network which are not in $V$ are called Steiner points (those in $V$ will be called regular points). A shortest tree interconnecting $V$ without using any Steiner points is called a minimal tree. Let $\sigma (V)$ and $\mu (V)$ denote the lengths of a Steiner minimal tree and a minimal tree, respectively. Define $\rho$ to be the greatest lower bound for the ratio $\sigma (V)/\mu (V)$ over all $V$. We prove $\rho > .8$.