  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.
