Abstract

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.

Keywords: Normed space; Gromov-Hausdorff distance; equilateral dimension.

MSC: 46B20, 46B85, 51F99

DOI: 10.57016/MV-JZAC3144

Pages:  1--9