Analytic formulas for full steiner trees

Abstract

It was conjectured by Gilbert and Pollak [6] that, for any finite set of points in the Euclidean plane, the ratio of the length of a Steiner minimal tree to the length of a minimal spanning tree is at least {Mathematical expression}. To date, this has been proved only for at most five points. In this paper, some analytic formulas for the length of full Steiner trees are considered. These provide an alternative proof of the conjecture for quadrilaterals, and the foundation for a possible approach for more complicated polygons. © 1991 Springer-Verlag New York Inc.