Publications de l'Institut Mathématique, Nouvelle Série Vol. 94(108), pp. 3–15 (2013) 

GROMOV MINIMAL FILLINGS FOR FINITE METRIC SPACESAlexander O. Ivanov, Alexey A. TuzhilinChair of Differential Geometry and Applications, Department of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, RussiaAbstract: The problem discussed in this paper was stated by Alexander O. Ivanov and Alexey A. Tuzhilin in 2009. It stands at the intersection of the theories of Gromov minimal fillings and Steiner minimal trees. Thus, it can be considered as onedimensional stratified version of Gromov minimal fillings problem. Here we state the problem; discuss various properties of onedimensional minimal fillings, including a formula calculating their weights in terms of some special metrics characteristics of the metric spaces they join (it was obtained by A. Yu. Eremin after many fruitful discussions with participants of Ivanov–Tuzhilin seminar in Moscow State University); show various examples illustrating how one can apply the developed theory to get nontrivial results; discuss the connection with additive spaces appearing in bioinformatics and classical Steiner minimal trees having many applications, say, in transportation problem, chip design, evolution theory etc. In particular, we generalize the concept of Steiner ratio and get a few its modifications defined by means of minimal fillings, which could give a new approach to attack the long standing Gilbert–Pollack Conjecture on the Steiner ratio of the Euclidean plane. Keywords: finite metric spaces, Gromov minimal fillings, Steiner minimal trees, extreme networks, Steiner ratio, additive spaces Classification (MSC2000): 51F99; 51K99 Full text of the article: (for faster download, first choose a mirror)
Electronic fulltext finalized on: 8 Nov 2013. This page was last modified: 22 Nov 2013.
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