The visual angle metric and quasiregular maps

TL;DR

This paper explores the properties of the visual angle metric in quasiregular maps, establishing key lemmas, inequalities, and conditions under which bilipschitz maps preserve quasiconformality in convex domains.

Contribution

It introduces a Schwarz-type lemma for quasiregular maps involving the visual angle metric and links bilipschitz conditions in this metric to quasiconformality.

Findings

Proved a Schwarz-type lemma for quasiregular maps with the visual angle metric.

Established that bilipschitz maps in the visual angle metric are also bilipschitz in the hyperbolic metric for certain domains.

Derived inequalities relating the visual angle metric to other metrics like the distance ratio and quasihyperbolic metrics.

Abstract

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.