The Visual Angle Metric and Möbius Transformations

Abstract

A new similarity invariant metric $v_G$ is introduced. The visual angle metric $v_G$ is defined on a domain $G\subsetneq {{\mathbb {R}}}^n$ 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 ${\mathbb {B}}^n$ or the upper half space ${\mathbb {H}}^n$. We also obtain the sharp Lipschitz constant for a Möbius transformation $f: G\rightarrow G'$ between domains $G$ and $G'$ in ${{\mathbb {R}}}^n$ with respect to the metrics $v_G$ and $v_{G'}$. For instance, in the case $G=G'={\mathbb {B}}^n$ the result is sharp.