The visual angle metric and M\"obius transformations

TL;DR

This paper introduces a new similarity invariant metric called the visual angle metric, establishes sharp bounds relating it to hyperbolic metrics in specific domains, and determines the optimal Lipschitz constants for Möbius transformations with respect to this metric.

Contribution

It defines the visual angle metric, finds sharp bounds in key domains, and computes the best Lipschitz constants for Möbius transformations relative to this metric.

Findings

Sharp bounds for the visual angle metric in the unit ball and upper half space.

Exact Lipschitz constants for Möbius transformations in these domains.

The metric is a new similarity invariant with applications in geometric analysis.

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 M\"obius transformation $f: G\rightarrow G'$ between domains $G$ and $G'$ in $\Rn$ with respect to the metrics $v_G$ and $v_{G'}$. For instance, in the case $G=G'=\Bn$ the result is sharp.