Minimum Steiner Trees in Normed Planes

A minimum Steiner tree for a given setX of points is a network interconnecting the points ofX 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 domainD having nonempty interior. We show that if the boundary ofD 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 ratioρ(D) forD, and show that, for anyD, 0.623<ρ(D)<0.8686.