Metric stability of trees and tight spans

TL;DR

This paper establishes optimal stability bounds for metric trees and injective spaces, showing how Gromov-Hausdorff distances between these structures relate to those of their generating sets.

Contribution

It provides sharp stability estimates for metric trees and tight spans, extending isometric relations and bounding Gromov-Hausdorff distances in these contexts.

Findings

GH distance of metric trees spanned by subsets is bounded by the GH distance of the subsets

GH distance between injective hulls is at most twice the GH distance of the original spaces

Optimal extension results for roughly isometric relations between metric trees and injective spaces

Abstract

In this note, we prove optimal extension results for roughly isometric relations between metric (R-)trees and injective metric spaces. This yields sharp stability estimates, in terms of the Gromov-Hausdorff (GH) distance, for certain metric spanning constructions: The GH distance of two metric trees spanned by some subsets is smaller than or equal to the GH distance of these sets. The GH distance of the injective hulls, or tight spans, of two metric spaces is at most twice the GH distance between themselves.