A new metric associated with the domain boundary

In this paper, we introduce a new metric $\tilde{c}$ which is associated with the domain boundary for a Ptolemy space $(X,d)$. Moreover, we study the inclusion relation of the $\tilde{c}$ metric balls and some related hyperbolic type metric balls in subdomains of $\mathbb{R}^n$. In addition, we study distortion properties of Möbius transformations with respect to the $\tilde{c}$ metric in the unit ball and the quasiconformality of bilipschitz mappings in $\tilde{c}$ metric.