On quasi-inversions

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 Gehring and Väisälä (Acta Math 114:1–70,1965). 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.