zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The Steiner ratio: the current state. (Russian) Zbl 1228.05121

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 N in a metric space (X,ρ). 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 N. 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).]

MSC:
05C05Trees
05B40Packing; covering (combinatorics)
46B20Geometry and structure of normed linear spaces
52B55Computational aspects related to geometric convexity
68R10Graph theory in connection with computer science (including graph drawing)