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We consider the proper class of all metric spaces endowed with theGromov--Hausdorff distance. Its maximal subclasses, consisting of the spaces onfinite distance from each other, we call clouds. Multiplying all distances in ametric space by the same positive real number, we obtain a similaritytransformation of the Gromov--Hausdorff class. In our previous work, weobserved that with such a transformation, some clouds can jump to others. Tocharacterize the phenomenon, we studied the stabilizers of the similarityaction. In this paper, we prove that every cloud with a nontrivial stabilizerhas a center, i.e., a metric space for which all similarities from thestabilizer generate a new space at zero distance. Moreover, the center isunique modulo zero distance. The proof is based on the cloud completenesstheorem.

Clouds in Gromov-Hausdorff Class: their completeness and centers