Dimension-free inner uniform estimates for quasigeodesics

Abstract

In this paper, we 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. Väisälä 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].

26 pages. The paper will not be submitted for publication