The Apple Doesn't Fall Far From the (Metric) Tree: The Equivalence of Definitions

TL;DR

This paper proves the equivalence of different definitions of metric trees and -hyperbolic spaces, enhancing understanding of their geometric properties and relationships to CAT() spaces.

Contribution

It establishes the equivalence of various definitions for metric trees and -hyperbolic spaces, clarifying their geometric and metric relationships.

Findings

Definitions of metric trees are equivalent.

Definitions of -hyperbolic spaces are equivalent.

Connections to CAT() spaces are clarified.

Abstract

In this paper we prove the equivalence of definitions for metric trees and for \delta-hyperbolic spaces. We point out how these equivalences can be used to understand the geometric and metric properties of \delta-hyperbolic spaces and its relation to CAT(\kappa) spaces.