Gromov hyperbolicity III: an improved geometric characterization and its applications

In the seminal work of Balogh-Buckley [Invent. Math. 2003], the authors asked the following fundamental open problem: for proper subdomains in the Euclidean space $\mathbb{R}^n$, does the ball separation condition alone imply the Gehring-Hayman inequality? In this paper, via a completely new measure-independent approach, we establish the following geometric characterization of Gromov hyperbolicity in a fairly general setting: The Gromov hyperbolicity of a proper subdomain in a doubling metric space is quantitatively equivalent to the geometric ball separation condition, with explicit dependence on the coefficients. In the special case of Euclidean spaces, it affirmatively solves the above Balogh-Buckely problem. Our result also significantly improves the main result of Koskela-Lammi-Manojlovi\'c [Ann. Sci. \'Ec. Norm. Sup\'er. 2014]. As applications, we obtain the quasiconformal invariance of ball separation condition, a geometric characterization of inner uniformity in terms of ball separation condition, and the Gromov hyperbolicity of quasihyperbolic John length spaces.