Gromov-Hausdorff distances between normed spaces

TL;DR

This paper investigates the Gromov-Hausdorff distance between finite-dimensional real normed spaces, establishing conditions for isometry and demonstrating that most spaces are infinitely distant from others in this metric.

Contribution

It proves that finite-dimensional normed spaces are isometric if their Gromov-Hausdorff distance is finite and shows that most such spaces are infinitely distant from non-isometric ones.

Findings

Finite-dimensional normed spaces with finite Gromov-Hausdorff distance are isometric.

Most finite-dimensional normed spaces are infinitely distant from all non-isometric spaces.

The paper extends understanding of the metric structure of normed spaces.

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.