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arXiv · 1208.2871

The visual angle metric and M\"obius transformations

Riku Klén·Henri Lindén·Matti Vuorinen·Gendi Wang
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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.

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Riku Klén, Henri Lindén, Matti Vuorinen, Gendi Wang. 2012-08-14. The visual angle metric and M\"obius transformations. https://arxiv.org/abs/1208.2871

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