  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.
