# Gromov hyperbolicity II: Dimension-free inner uniform estimates for quasigeodesics

## TL;DR

This paper establishes a dimension-free inner uniform estimate for quasigeodesics in Gromov hyperbolic spaces, extending results to infinite-dimensional Banach spaces and resolving several open problems in geometric analysis.

## Contribution

It proves that quasigeodesics in Gromov hyperbolic John domains are inner uniform with constants independent of dimension, generalizing previous results and answering open questions.

## Key findings

- Dimension-free inner uniform estimates for quasigeodesics.
- Extension of results to general Banach spaces.
- Resolution of open problems by Heinonen and V"ais"al"a.

## Abstract

This is the second article of a series of our recent works, addressing an open question of Bonk-Heinonen-Koskela [3], to study the relationship between (inner) uniformality and Gromov hyperbolicity in infinite dimensional spaces. Our main focus of this paper is to establish a dimension-free inner uniform estimate for quasigeodesics. More precisely, we prove that a $c_0$-quasigeodesic in a $\delta$-Gromov hyperbolic $c$-John domain in $\mathbb{R}^n$ is $b$-inner uniform, for some constant $b$ depending only on $c_0$, $\delta$ and $c$, but not on the dimension $n$. The proof relies crucially on the techniques introduced by Guo-Huang-Wang in their recent work [arXiv:2502.02930, 2025]. In particular, we actually show that the above result holds in general Banach spaces, which answers affirmatively an open question of J. V\"ais\"al\"a in [Analysis, 2004] and partially addresses the open question of Bonk-Heinonen-Koskela in [Asterisque, 2001]. As a byproduct of our main result, we obtain that a $c_0$-quasigeodesic in a $\delta$-Gromov hyperbolic $c$-John domain in $\mathbb{R}^n$ is a $b$-cone arc with a dimension-free constant $b=b(c_0,\delta,c)$. This resolves an open problem of J. Heinonen in [Rev. Math. Iberoam., 1989].