Steiner Problem in the Gromov-Hausdorff Space: The Case of Finite Metric Spacesстатья
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Аннотация:Steiner Problem in the Gromov--Hausdorff space, i.e., in the space of compact metric spaces (considered up to an isometry) endowed with the Gromov--Hausdorff distance is considered. Since this space is not proper, the existence problem for shortest networks in this space is open. It is shown that each finite family of finite metric spaces can be connected by a shortest tree. Moreover, it turns out that in this case there exists a shortest tree such that all its vertices are finite metric spaces. An estimate for the number of points in those metric spaces is obtained. As an example, the case of three-point metric spaces is considered. It is also shown that the Gromov--Hausdorff space does not realise minimal fillings, i.e., shortest trees in this space need not be minimal fillings of their boundaries.