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.
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
\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{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}
Linking options:
https://www.mathnet.ru/eng/timm1475
https://www.mathnet.ru/eng/timm/v23/i4/p152
This publication is cited in the following 2 articles:
A. Kh. Galstyan, “Problema Ferma—Shteinera v prostranstve kompaktnykh podmnozhestv evklidovoi ploskosti”, Materialy XVII Vserossiiskoi molodezhnoi shkoly-konferentsii «Lobachevskie chteniya-2018», 23-28 noyabrya 2018 g., Kazan. Chast 1, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 175, VINITI RAN, M., 2020, 44–55
O. S. Malysheva, “Optimal position of compact sets and the Steiner problem in spaces with Euclidean Gromov-Hausdorff metric”, Sb. Math., 211:10 (2020), 1382–1398
Citation in format AMSBIB
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