This paper is a survey of recent results on the study of the Steiner ratio of metric spaces, in particular Riemannian manifolds and normed spaces. Some of these results belong to the authors (see, for example, [A.O. Ivanov, A.A. Tuzhilin and D. Cieslik, Math. Notes 74, No. 3, 367–374; translation from Mat. Zametki 74, No. 3, 387–395 (2003; Zbl 1066.52009)]).
The Steiner ratio can be considered as the characteristic of ‘goodness’ of approximated solutions to the well-known Steiner problem, which searches for a network of minimal length that spans a finite set of points in a metric space (). This shortest network has to be a tree and is called the Steiner minimum tree. The Steiner ratio is the greatest lower bound for the ratio of the length of the Steiner minimum tree to the length of the minimum spanning tree for the given point set . The interest in approximated solutions to the Steiner problem is due, on the one hand, to the wealth of applications of the shortest networks, and, on the other hand, to the fact that the Steiner problem is NP-complete.
The authors start with necessary definitions and proceed with the estimates and exact values of the Steiner ratio for general metric spaces and Riemannian manifolds. Then they discuss the problems related to the Steiner ratio in normed spaces. At the end of the paper some relations between the Steiner ratio and some known problems of discrete geometry are discussed. For example, the authors show how the Steiner ratio can be used in the study of packing and covering problems in Euclidean space. Some open problems are also listed and a comprehensive bibliography is given.
[Reviewed by Lyuba S. Alboul (MR2269103).]