On Quasi-inversions

TL;DR

This paper investigates the properties of quasi-inversions in starlike domains, establishing conditions under which they are bi-Lipschitz and exploring their relation to tangent conditions and boundary parametrizations.

Contribution

It introduces the concept of quasi-inversion in starlike domains and characterizes when it is bi-Lipschitz using the $ ext{alpha}$-tangent condition and polar parametrization.

Findings

Quasi-inversion is bi-Lipschitz iff tangent lines are far from the origin.

Bi-Lipschitz constant approaches 1 as boundary approaches the unit sphere.

Polar parametrization is bi-Lipschitz iff the domain satisfies the $ ext{alpha}$-tangent condition.

Abstract

Given a bounded domain $D \subset {\mathbb R}^n$ strictly starlike with respect to $0 \in D\,,$ we define a quasi-inversion w.r.t. the boundary $\partial D \,.$ We show that the quasi-inversion is bi-Lipschitz w.r.t. the chordal metric if and only if every "tangent line" of $\partial D$ is far away from the origin. Moreover, the bi-Lipschitz constant tends to $1,$ when $\partial D$ approaches the unit sphere in a suitable way. For the formulation of our results we use the concept of the $\alpha$-tangent condition due to F. W. Gehring and J. V\"ais\"al\"a (Acta Math. 1965). This condition is shown to be equivalent to the bi-Lipschitz and quasiconformal extension property of what we call the polar parametrization of $\partial D$. In addition, we show that the polar parametrization, which is a mapping of the unit sphere onto $\partial D\,,$ is bi-Lipschitz if and only if $D$ satisfies the $\alpha$-tangent condition.