Steiner's problem in the Gromov--Hausdorff space: the case of finite metric spaces
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 4, pp. 152-161
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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.
@article{TIMM_2017_23_4_a14,
author = {A. O. Ivanov and N. K. Nikolaeva and A. A. Tuzhilin},
title = {Steiner's problem in the {Gromov--Hausdorff} space: the case of finite metric spaces},
journal = {Trudy Instituta matematiki i mehaniki},
pages = {152--161},
publisher = {mathdoc},
volume = {23},
number = {4},
year = {2017},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TIMM_2017_23_4_a14/}
}
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A. O. Ivanov; N. K. Nikolaeva; A. A. Tuzhilin. Steiner's problem in the Gromov--Hausdorff space: the case of finite metric spaces. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 4, pp. 152-161. http://geodesic.mathdoc.fr/item/TIMM_2017_23_4_a14/