On stars and Steiner stars

A Steiner star   for a set PP of nn points in RdRd connects an arbitrary point in RdRd to all points of PP, while a star   connects one of the points in PP to the remaining n−1n−1 points of PP. All connections are realized by straight line segments. Let the length   of a graph be the total Euclidean length of its edges. Fekete and Meijer showed that the minimum star is at most 2 times longer than the minimum Steiner star for any finite point configuration in RdRd. The supremum of the ratio between the two lengths, over all finite point configurations in RdRd, is called the star Steiner ratio   in RdRd. It is conjectured that this ratio is 4/π=1.2732…4/π=1.2732… in the plane and 4/3=1.3333…4/3=1.3333… in three dimensions. Here we give upper bounds of 1.3631 in the plane, and 1.3833 in 3-space. These estimates yield improved upper bounds on the maximum ratios between the minimum star and the maximum matching in two and three dimensions. We also verify that the conjectured bound 4/π4/π in the plane holds in two special cases.Our method exploits the connection with the classical problem of estimating the maximum sum of pairwise distances among nn points on the unit sphere, first studied by László Fejes Tóth. It is quite general and yields the first nontrivial estimates below 2 on the star Steiner ratios in arbitrary dimensions. We show, however, that the star Steiner ratio in RdRd tends to 2 as dd goes to infinity. As it turns out, our estimates are related to the classical infinite Wallis product: π2=∏n=1∞(4n24n2−1)=21⋅23⋅43⋅45⋅65⋅67⋅87⋅89⋯.