Geodesic
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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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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/

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