Optimal position of compact sets and the Steiner problem in spaces with Euclidean Gromov-Hausdorff metric

We study the geometry of the metric space of compact subsets of $\mathbb R^n$ considered up to an orientation-preserving motion. We show that, in the optimal position of a pair of compact sets (for which the Hausdorff distance between the sets cannot be decreased), one of which is a singleton, this point is at the Chebyshev centre of the other. For orientedly similar compacta we evaluate the Euclidean Gromov-Hausdorff distance between them and prove that, in the optimal position, the Chebyshev centres of these compacta coincide. We show that every three-point metric space can be embedded isometrically in the space of compacta under consideration. We prove that, for a pair of optimally positioned compacta all compacta that lie in between in the sense of the Hausdorff metric also lie in between in the sense of the Euclidean Gromov-Hausdorff metric. For an arbitrary $n$-point boundary formed by compact sets of a set $\mathscr X$ that are neighbourhoods of segments, the Steiner point realizes the minimal filling and also belongs to the set $\mathscr X$. Bibliography: 14 titles.

Steiner point, Euclidean Gromov-Hausdorff metric, optimal position of compacta