Extendability of Metric Segments in Gromov--Hausdorff Distance

TL;DR

This paper investigates the geometric properties of the Gromov-Hausdorff distance on metric spaces, focusing on the intrinsic nature of this distance, the structure of continuous curves, and the extendability of metric segments.

Contribution

It introduces the concept of continuous curves and their lengths in the Gromov-Hausdorff space, proving the intrinsic nature of the distance and analyzing the extension of metric segments.

Findings

Gromov-Hausdorff distance is shown to be intrinsic.

Continuous curves and their lengths are defined in the space.

Extension of metric segments beyond endpoints is studied.

Abstract

In this paper geometry of Gromov-Hausdorff distance on the class of all metric spaces considered up to an isometry is investigated. For this class continuous curves and their lengths are defined, and it is shown that the Gromov-Hausdorff distance is intrinsic. Besides, metric segments are considered, i.e., the classes of points lying between two given ones, and an extension problem of such segments beyond their end-points is considered.