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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.
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
Linking options:
https://www.mathnet.ru/eng/timm1475 https://www.mathnet.ru/eng/timm/v23/i4/p152
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