On the Steiner ratio in 3-space

The “Steiner minimal tree” (SMT) of a point set P is the shortest network of “wires” which will suffice to electrically interconnnect P. The “minimum spanning tree” (MST) is the shortest such network when only intersite line segments are permitted. The “Steiner ratio” ρ(P) of a point set P is the length of its SMT divided by the length of its MST. It is of interest to understand which point set (or point sets) in Rd has minimal Steiner ratio. In this paper, we introduce a point set in Rd which we call the “d-dimensional sausage.” The one- and two-dimensional sausages have minimal Steiner ratios 1 and √32, respectively. (The 2-sausage is the vertex set of an infinite strip of abutting equilateral triangles. The 3-sausage is an infinite number of points evenly spaced along a helix.) We present extensive heuristic evidence to support the conjecture that the 3-sausage also has minimal Steiner ratio (≈ 0.784190373377122). Also, we prove that the regular tetrahedron minimizes ρ among 4-point sets to at least 12 decimal places of accuracy. This is a companion paper to D-Z. Du and W. D. Smith, “Three Disproofs of the Gilbert-Pollak Steiner Ratio Conjecture in Three or More Dimensions,” to be published in the Journal of Combinatorial Theory. We have tried to devote this paper more to 3D and the other paper more to general dimensions, but the split is not clean.