The visual angle metric and M\"obius transformations

A new similarity invariant metric $v_G$ is introduced. The visual angle metric $v_G$ is defined on a domain $G⊊\text{Rn}$ whose boundary is not a proper subset of a line. We find sharp bounds for $v_G$ in terms of the hyperbolic metric in the particular case when the domain is either the unit ball $\text{Bn}$ or the upper half space $\text{Hn}$. We also obtain the sharp Lipschitz constant for a M"obius transformation $f:G\to G^{\mathrm{\prime }}$ between domains $G$ and $G^{\mathrm{\prime }}$ in $\text{Rn}$ with respect to the metrics $v_G$ and $v_{G^{\mathrm{\prime }}}$. For instance, in the case $G=G^{\mathrm{\prime }}=\text{Bn}$ the result is sharp.