Clouds in Gromov–Hausdorff class: Their completeness and centers

摘要

? 2022We consider the proper class of all metric spaces endowed with the Gromov–Hausdorff distance. Its maximal subclasses, consisting of spaces at finite distance from each other, we call clouds. Multiplying all distances in a metric space by the same positive real number, we obtain a similarity transformation of the Gromov–Hausdorff class. In our previous work, we observed that with such a transformation, some clouds can jump to others. To characterize the phenomenon, we studied the stabilizers of the similarity action. In this paper, we prove that every cloud with a nontrivial stabilizer has a center, i.e., a metric space for which all similarities from the stabilizer generate a new space at zero distance. Moreover, the center is unique modulo zero distance. The proof is based on the cloud completeness theorem.