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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2017, Volume 23, Number 4, Pages 152–161
DOI: https://doi.org/10.21538/0134-4889-2017-23-4-152-161
(Mi timm1475)
 

This article is cited in 2 scientific papers (total in 2 papers)

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
Full-text PDF (221 kB) Citations (2)
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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.
Received: 23.06.2017
English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2019, Volume 304, Issue 1, Pages S88–S96
DOI: https://doi.org/10.1134/S008154381902010X
Bibliographic databases:
Document Type: Article
UDC: 514+519.1
Language: Russian
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; Proc. Steklov Inst. Math. (Suppl.), 304, suppl. 1 (2019), S88–S96
Citation in format AMSBIB
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\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{https://elibrary.ru/item.asp?id=30713969}
\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2019
\vol 304
\issue , suppl. 1
\pages S88--S96
\crossref{https://doi.org/10.1134/S008154381902010X}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000453521700014}
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  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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