Local Structure of Gromov-Hausdorff Space near Finite Metric Spaces in General Position

Abstract

We investigate the local structure of the space $\mathcal{M}$ consisting of isometry classes of compact metric spaces, endowed with the Gromov-Hausdorff metric. We consider finite metric spaces of the same cardinality and suppose that these spaces are in general position, i.e., all nonzero distances in each of the spaces are distinct, and all triangle inequalities are strict. We show that sufficiently small balls in $\mathcal{M}$ centered at these spaces and having the same radii are isometric. As consequences, we prove that the cones over such spaces (with the vertices at one-point space) are isometrical; the isometry group of each sufficiently small ball centered at a general position $n$-points space, $n\ge3$, contains a subgroup isomorphic to the group $S_n$ of permutations of a set containing $n$ points.