Examples of hyperbolic spaces without the properties of quasi-ball or bounded eccentricity

TL;DR

This paper provides examples of hyperbolic metric spaces that do not exhibit quasi-ball properties or bounded eccentricity in ball intersections, answering an open question in the field.

Contribution

It constructs specific hyperbolic spaces lacking quasi-ball and bounded eccentricity properties, challenging previous assumptions.

Findings

Examples of hyperbolic spaces without quasi-ball property

Intersections of metric balls do not resemble balls or have bounded eccentricity

Addresses an open question in hyperbolic geometry

Abstract

In this note, we present examples of non-quasi-geodesic metric spaces which are hyperbolic (i.e., satisfying the Gromov's $4$-point condition) while the intersection of any two metric balls therein does not either "look like" a ball or has uniformly bounded eccentricity. This answers an open question posed in [2].