  We establish a version of a classical theorem of Pommerenke, which is a
diameter version of the Gehring-Hayman inequality on Gromov hyperbolic domains
of $\mathbb{R}^n$. Two applications are given. Firstly, we generalize
Ostrowski's Faltensatz to quasihyperbolic geodesics of Gromov hyperbolic
domains. Secondly, we prove that unbounded uniform domains can be characterized
in the terms of Gromov hyperbolicity and a naturally quasisymmetric
correspondence on the boundary, where the Gromov boundary is equipped with a
Hamenst\"adt metric (defined by using a Busemann function).
