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 Trudy Inst. Mat. i Mekh. UrO RAN, 2017, Volume 23, Number 4, Pages 152–161 (Mi timm1475)

Steiner's problem in the Gromov–Hausdorff space: the case of finite metric spaces

A. O. Ivanova, N. K. Nikolaevab, A. A. Tuzhilina

a Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Moscow, 119991 Russia
b SOSh NOU “Orthodox Saint-Peter School”, Moscow, 109028, Tessinskiy per., 3 Russia

Abstract: We study Steiner's problem in the Gromov–Hausdorff space, i.e., in the space of compact metric spaces (considered up to isometry) endowed with the Gromov-Hausdorff distance. Since this space is not boundedly compact, the problem of the existence of a shortest network connecting a finite point set in this space is open. We prove that each finite family of finite metric spaces can be connected by a shortest network. Moreover, it turns out that there exists a shortest tree all of whose vertices are finite metric spaces. A bound for the number of points in such metric spaces is derived. As an example, the case of three-point metric spaces is considered. We also prove that the Gromov-Hausdorff space does not realise minimal fillings, i.e., shortest trees in it need not be minimal fillings of their boundaries.

Keywords: Steiner's problem, shortest network, Steiner's minimal tree, minimal filling, Gromov-Hausdorff space, Gromov–Hausdorff distance.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00378 Ministry of Education and Science of the Russian Federation ÍØ-7962.2016.1

DOI: https://doi.org/10.21538/0134-4889-2017-23-4-152-161

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Bibliographic databases:

UDC: 514+519.1
MSC: 58E10, 49K35, 05C35, 05C10, 30L05
Received: 23.06.2017

Citation: A. O. Ivanov, N. K. Nikolaeva, A. A. Tuzhilin, “Steiner's problem in the Gromov–Hausdorff space: the case of finite metric spaces”, Trudy Inst. Mat. i Mekh. UrO RAN, 23, no. 4, 2017, 152–161

Citation in format AMSBIB
\Bibitem{IvaNikTuz17} \by A.~O.~Ivanov, N.~K.~Nikolaeva, A.~A.~Tuzhilin \paper Steiner's problem in the Gromov--Hausdorff space: the case of finite metric spaces \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2017 \vol 23 \issue 4 \pages 152--161 \mathnet{http://mi.mathnet.ru/timm1475} \crossref{https://doi.org/10.21538/0134-4889-2017-23-4-152-161} \elib{http://elibrary.ru/item.asp?id=30713969} 

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