On the uniqueness of quasihyperbolic geodesics

TL;DR

This paper addresses open problems in quasihyperbolic geometry, proving geodesic uniqueness in planar domains and convexity of small quasihyperbolic balls in higher dimensions, advancing understanding of metric properties.

Contribution

It establishes geodesic uniqueness in simply connected planar domains and for points within a quasihyperbolic distance less than π, and proves convexity of small quasihyperbolic balls in higher dimensions.

Findings

Quasihyperbolic geodesics are unique in simply connected planar domains.

Geodesics are unique for pairs of points with quasihyperbolic distance less than π.

Small quasihyperbolic balls are convex in all dimensions.

Abstract

In this work we solve a couple of well known open problems related to the quasihyperbolic metric. In the case of planar domains, our first main result states that quasihyperbolic geodesics are unique in simply connected domains. As the second main result, we prove that for an arbitrary plane domain the geodesics are unique for all pairs of points with quasihyperbolic distance less than $\pi$. This bound is sharp and improves the best known bound. Concerning the $n$-dimensional case, we prove the existence of a universal constant $c$ (independent of dimension) such that any quasihyperbolic ball with radius less than $c$ is convex.