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Convex Regions in the Plane and their Domes

Authors:	Epstein, D. B. A. Marden, A. Markovic, V.

Affiliation:	Mathematics Institute, University of Warwick Coventry, CV4 7AL, United Kingdom dbae{at}maths.warwick.ac.uk, markovic{at}maths.warwick.ac.uk Mathematics Department, University of Minnesota Minneapolis, MN 55455, USA am{at}math.umn.edu

Abstract:	We make a detailed study of the relation of a euclidean convexregion ${Omega}$ $sub$ C $->$ Dome ( ${Omega}$ ). The dome is the relative boundary, in theupper halfspace model of hyperbolic space, of the hyperbolicconvex hull of the complement of ${Omega}$ . The first result is to provethat the nearest point retraction r: ${Omega}$ $->$ Dome ( ${Omega}$ ) is 2-quasiconformal.The second is to establish precise estimates of the distortionof r near ${partial}$ ${Omega}$ . 2000 Mathematics Subject Classification 30C75,30F40, 30F45, 30F60.

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