Local geometry of the Gromov–Hausdorff metric space and totally asymmetric finite metric spaces

In the present paper, we investigate the structure of the metric space $\mathcal M$ of compact metric spaces considered up to an isometry and endowed with the Gromov–Hausdorff metric in a neighbourhood of a finite metric space, whose isometry group is trivial. It is shown that a sufficiently small ball in the subspace of $\mathcal M$ consisting of finite spaces with the same number of points centered at such a space is isometric to a corresponding ball in the space $\mathbb R^N$ endowed with the norm $|(x_1, \dots, x_N ) | = \max\limits_{i} |x_i|$. Also an isometric embedding of a finite metric space into a neighbourhood of a finite asymmetric space in $\mathcal M$ is constructed.