2022/02/15 by S. I. Bogataya, S. A. Bogatyy, Bogataya, S. I. +5 · 3 citations
Mathematics · #51F99 #Advanced Algebra and Geometry #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2202.07337
openalex publication_date 2022/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the proper class of all metric spaces endowed with the Gromov--Hausdorff distance. Its maximal subclasses, consisting of the spaces on 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.