The Nikolov–Andreev Metric and Gromov Hyperbolicity

In this paper, we prove that a proper subdomain \(\Omega \) $\mathrm{\Omega }$ of \(\mathbb {R}^n\) ${\mathbb{R}}^n$ equipped with the metric \(i_{\Omega }\) $i_{\mathrm{\Omega }}$ , recently introduced by Nikolov and Andreev, is Gromov hyperbolic. We also show that there is a natural quasisymmetric correspondence between the Euclidean boundary of \(\Omega \) $\mathrm{\Omega }$ (with respect to \(\overline{\mathbb {R}^n}\) $\overline{{\mathbb{R}}^n}$ ) and the Gromov boundary of \((\Omega ,i_\Omega )\) $(\mathrm{\Omega },i_{\mathrm{\Omega }})$ .