On stars and Steiner stars. II

TL;DR

This paper investigates the ratio between the lengths of minimum stars and Steiner stars in Euclidean spaces, providing improved upper bounds and insights into related geometric optimization problems.

Contribution

It establishes new upper bounds for the star Steiner ratio in the plane and three-dimensional space, improving upon previous estimates.

Findings

Upper bound of 1.3631 in the plane

Upper bound of 1.3833 in 3-space

Enhanced bounds on ratios between minimum star and maximum matching

Abstract

A {\em Steiner star} for a set $P$ of $n$ points in $\RR^d$ connects an arbitrary center point to all points of $P$, while a {\em star} connects a point $p\in P$ to the remaining $n-1$ points of $P$. All connections are realized by straight line segments. Fekete and Meijer showed that the minimum star is at most $\sqrt{2}$ times longer than the minimum Steiner star for any finite point configuration in $\RR^d$. The maximum ratio between them, over all finite point configurations in $\RR^d$, is called the {\em star Steiner ratio} in $\RR^d$. It is conjectured that this ratio is $4/\pi = 1.2732...$ in the plane and $4/3=1.3333...$ in three dimensions. Here we give upper bounds of 1.3631 in the plane, and 1.3833 in 3-space, thereby substantially improving recent upper bounds of 1.3999, and $\sqrt{2}-10^{-4}$, respectively. Our results also imply improved bounds on the maximum ratios between the minimum star and the maximum matching in two and three dimensions.