This article is cited in 49 papers

One-dimensional Gromov minimal filling problem

A. O. Ivanov^ab, A. A. Tuzhilin<sup>ba

a</sup> M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
^b Laboratory of Discrete and Computational Geometry named after B. N. Delone of P. G. Demidov Yaroslavl State University

Abstract: The paper is devoted to a new branch in the theory of one-dimensional variational problems with branching extremals, the investigation of one-dimensional minimal fillings introduced by the authors. On the one hand, this problem is a one-dimensional version of a generalization of Gromov's minimal fillings problem to the case of stratified manifolds. On the other hand, this problem is interesting in itself and also can be considered as a generalization of another classical problem, the Steiner problem on the construction of a shortest network connecting a given set of terminals. Besides the statement of the problem, we discuss several properties of the minimal fillings and state several conjectures.
Bibliography: 38 titles.

Keywords: metric spaces, Gromov minimal fillings, Steiner minimal trees, minimal spanning trees, Steiner ratio.

UDC: 514.774.8+515.124.4+519.176

MSC: Primary 05C12, 54E35; Secondary 05C05, 52A38

Received: 04.08.2010 and 08.04.2011

DOI: 10.4213/sm7777

 English version: Sbornik: Mathematics, 2012, 203:5, 677–726

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