  In the present paper we study the original Gromov-Hausdorff distance between
real normed spaces. In the first part of the paper we prove that two
finite-dimensional real normed spaces on a finite Gromov-Hausdorff distance are
isometric to each other. We then study the properties of finite point sets in
finite-dimensional normed spaces whose cardinalities exceed the equilateral
dimension of an ambient space. By means of the obtained results we prove the
following enhancement of the aforementioned theorem: every finite-dimensional
normed space lies on an infinite Gromov-Hausdorff distance from all other
non-isometric normed spaces.
