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Trudy Inst. Mat. i Mekh. UrO RAN, 2017, Volume 23, Number 4, Pages 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

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 -7962.2016.1


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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2019, 304, suppl. 1, S88–S96

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; Proc. Steklov Inst. Math. (Suppl.), 304, suppl. 1 (2019), S88–S96

Citation in format AMSBIB
\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
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2019
\vol 304
\issue , suppl. 1
\pages S88--S96

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    This publication is cited in the following articles:
    1. A. Kh. Galstyan, “Problema Ferma—Shteinera v prostranstve kompaktnykh podmnozhestv evklidovoi ploskosti”, MaterialyXVII Vserossiiskoi molodezhnoishkoly-konferentsii Lobachevskie chteniya-2018,23-28 noyabrya 2018 g., Kazan.Chast 1, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 175, VINITI RAN, M., 2020, 44–55  mathnet  crossref
    2. O. S. Malysheva, “Optimalnoe polozhenie kompaktov i problema Shteinera v prostranstvakh s evklidovoi metrikoi Gromova–Khausdorfa”, Matem. sb., 211:10 (2020), 32–49  mathnet  crossref
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