Optimal location of compact in spaces with Euclidean invariant Gromov–Hausdorff metrics

We study nonempty compact subsets of the Euclidean space disposed optimally (the Hausdorff distance between them cannot be reduced). We show that if one of them is a singleton, then it coincides with the Chebyshev center of the second one. We also consider many other particular cases. As an application, we show that each three-point metric space can be isometrically embedded into the orbits space of the group of proper motions acting on the compact subsets of the Euclidean space. In addition, we prove that for each couple of optimally located compacts, all compacts intermediate in the sense of Hausdorff metric are intermediate in the sense of Euclidean Gromov–Hausdorff metric too.