Local Geometry of the Gromov–Hausdorff Metric Space and Totally Asymmetric Finite Metric Spaces

Abstract

In the present paper, we investigate the structure of the metric space M of compact metric spaces considered up to an isometry and endowed with the Gromov–Hausdorff metric in a neighborhood of a finite metric space, whose isometry group is trivial. It is shown that a sufficiently small ball in the subspace of M consisting of finite spaces with the same number of points centered at such a space is isometric to a corresponding ball in the space ℝN endowed with the norm |(x1, . . . , xN)| = \( \underset{i}{\max}\left|{x}_i\right| \). Also an isometric embedding of a finite metric space into a neighborhood of a finite asymmetric space in M is constructed.