Improved upper bounds for the Steiner ratio

Let P={P1,P2,…,Pn} be a set of nn points in RdRd. For every 1≤i≤n1≤i≤n, define the star rooted at  PiPi as the union of all straight line segments joining PiPi to all the other points in the set PP. A Steiner star   is the union of all straight line segments connecting some point in RdRd to each point of PP. The length of a star is defined as the total Euclidean length of its edges. We consider the problem of estimating the supremum of the ratio between the rooted star of minimal length and the Steiner star of minimal length, taken over all nn point configurations in RdRd. This is known as the Steiner ratio   in RdRd. It is conjectured that this ratio is 4/π4/π when d=2d=2 and 4/34/3 when d=3d=3. Fekete and Meijer proved that for every dd, this ratio is bounded from above by 2. Very recently, Dumitrescu, Tóth and Xu proved better upper bounds: 1.3631 for d=2d=2 and 1.3833 for d=3d=3. By a refinement of their approach we further improve these bounds to 1.3546 in the plane and 1.3801 in 3-space. These estimates yield improved upper bounds on the maximum ratio between the minimum star and the maximum matching in two and three dimensions.