2012/12/04 by David Kalaj, Kalaj, David, Матти Вуоринен +3
Mathematics · Medicine · #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Pelvic and Acetabular Injuries
paper · pdf · doi:10.48550/arxiv.1212.0721
openalex publication_date 2012/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a bounded domain D ⊂ \mathbb 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.