Gromov-Hausdorff Geometry of Metric Trees

TL;DR

This paper explores the Gromov-Hausdorff geometry of metric trees, establishing conditions under which Hausdorff and Gromov-Hausdorff distances coincide, and introduces new shortest geodesics connecting subsets to the entire tree.

Contribution

It provides a novel condition for the equality of Hausdorff and Gromov-Hausdorff distances in metric trees and constructs new shortest geodesics in the space of all metric spaces.

Findings

Condition for Hausdorff and Gromov-Hausdorff distances to coincide

Construction of new shortest geodesics in metric trees

Application demonstrated on subsets of the real line

Abstract

In this paper, we study metric trees, without any finiteness restrictions. For subsets of such trees, a condition that guarantees that the Hausdorff and Gromov--Hausdorff distances from the subset to the entire metric tree are the same is obtained. This result allows to construct a new class of shortest geodesics (in the proper class of all metric spaces) connecting such subset of a metric tree with the tree itself. In particular, the technique elaborated is demonstrated on subsets of the real line.