Minimum steiner trees in normed planes

Abstract

A minimum Steiner tree for a given set X of points is a network interconnecting the points of X having minimum possible total length. In this note we investigate various properties of minimum Steiner trees in normed planes, i.e., where the "unit disk" is an arbitrary compact convex centrally symmetric domain D having nonempty interior. We show that if the boundary of D is strictly convex and differentiable, then each edge of a full minimum Steiner tree is in one of three fixed directions. We also investigate the Steiner ratio ρ(D) for D, and show that, for any D, 0.623 <0.8686. © 1993 Springer-Verlag New York Inc.