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Register a new userThe visual angle metric and quasiregular maps
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The distortion of distances between points under maps is studied. We first prove a Schwarz-type lemma for quasiregular maps of the unit disk involving the visual angle metric. Then we investigate conversely the quasiconformality of a bilipschitz map with respect to the visual angle metric on convex domains. For the unit ball or half space, we prove that a bilipschitz map with respect to the visual angle metric is also bilipschitz with respect to the hyperbolic metric. We also obtain various inequalities relating the visual angle metric to other metrics such as the distance ratio metric and the quasihyperbolic metric.
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A new similarity invariant metric $v_G$ is introduced. The visual angle metric $v_G$ is defined on a domain $GsubsetneqRn$ 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 Mobius transformation $f: Grightarrow G$ 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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