On Quasi-inversions

Original · EN

Given a bounded domain D ⊂ Rⁿ strictly starlike with respect to 0 ∈ D, we define a quasi-inversion w.r.t. the boundary ∂ D. We show that the quasi-inversion is bi-Lipschitz w.r.t. the chordal metric if and only if every "tangent line" of ∂ D is far away from the origin. Moreover, the bi-Lipschitz constant tends to 1, when ∂ D approaches the unit sphere in a suitable way. For the formulation of our results we use the concept of the α-tangent condition due to F. W. Gehring and J. Väisälä (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 ∂ D. In addition, we show that the polar parametrization, which is a mapping of the unit sphere onto ∂ D, is bi-Lipschitz if and only if D satisfies the α-tangent condition.