Möbius Invariant Cassinian Metric

We introduce a new Mobius invariant, \(\delta \)-hyperbolic metric \(\tau _D\) for domains D in \(\overline{\mathbb R^n}\), which can be considered as a Mobius invariant analogue of the scale-invariant Cassinian metric \(\tilde{\tau }_D\) recently introduced by the author. We establish basic properties of \(\tau _D\) including its connections with \(\tilde{\tau }_D\), the Apollonian metric, Seittenranta’s metric and the hyperbolic metric. We also show that \(\tau _D\) is monotonic with respect to domains, its density is the same as the density of Ferrand’s metric and that the \(\tau _D\)-isometries of twice-punctured spaces are Mobius maps.