Convex Regions in the Plane and their Domes

Abstract

We make a detailed study of the relation of a euclidean convex region Ω ⊂ C → Dome (Ω). The dome is the relative boundary, in the upper halfspace model of hyperbolic space, of the hyperbolic convex hull of the complement of Ω. The first result is to prove that the nearest point retraction r: Ω → Dome (Ω) is 2-quasiconformal. The second is to establish precise estimates of the distortion of r near ∂ Ω.

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