A Dilemma on the Euclidean Steiner Ratio for helical points sets
R. P. Mondaini · 2004
An interesting dilemma is formulated in the search of the Steiner Ratio for helical point sets with a usual set topology. A proposal is made in order to circumvent this dilemma. In recent work [1],[2], we have introduced some analytical formulae for the study of the Steiner Ratio with Euclidean distance. We have also observed that our considerations can lead to a disproof of a famous conjecture which takes as the best upper value of the Steiner Ratio the value obtained from a set of vertices of regular tetrahedra glued together at common faces. However, we have been reluctant at announcing a disproof after we got some knowledge about specific restrictions of our point set configurations. These are derived from the fact that we can restrict ourselves to look for full Steiner Trees (FST). The dilemma can then be considered in the form: To each sequence s = 1, 2, 3, 4, ... given by j +1, 2j +1, 3j +1, 4j +1, ... with 0 ≤ j ≤ n−1, we can associate a minimum spanning tree (MST) and a Steiner minimal tree (SMT). The corresponding Steiner Ratio functions are given by the ratio of the lengths of these trees, for n →∞ or