Steiner Problem in the Gromov-Hausdorff Space: The Case of Finite Metric Spaces

Abstract

The Steiner problem is considered 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 in this space is open. It is shown that each finite family of finite metric spaces can be connected by a shortest network. Moreover, it turns out that in this case there exists a shortest tree all of whose vertices are finite metric spaces. An estimate for the number of points in these 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 realize minimal fillings; i.e., shortest trees in this space need not be minimal fillings of their boundaries.