To hyperbolize a metric space (X, d), we propose a new one-point hyperbolic metric \(\Gamma _p\) and investigate its average. We demonstrate the Gromov hyperbolicity of these metrics and compare the newly constructed metric \(\Gamma _p\) to several existing metrics including the metric d. We investigate the quasiconformality of the identity map \(\operatorname {id}: (X,d)\rightarrow (X, \Gamma _p)\) and analyze the relationship between the identity map and bilipschitzian mappings.
