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 $G\subsetneq\Rn$ 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 $\Bn$ or the upper half space $\Hn$. We also obtain the sharp Lipschitz constant for a Möbius transformation $f:G\to G^{\prime }$ between domains $G$ and $G^{\prime }$ in $\Rn$ with respect to the metrics $v_G$ and $v_{G^{\prime }}$. For instance, in the case $G=G'=\Bn$ the result is sharp.

25 pages, 13 figures

References in corpus (1)

• Weak metrics on Euclidean domains

Cited by in corpus (1)

• Comparison theorems for hyperbolic type metrics