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CONVEX REGIONS IN THE PLANE AND THEIR DOMES

Published online by Cambridge University Press:  18 April 2006

D. B. A. EPSTEIN ,
A. MARDEN  and
V. MARKOVIC
D. B. A. EPSTEIN
Affiliation:
Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdomdbae@maths.warwick.ac.uk, markovic@maths.warwick.ac.uk
A. MARDEN
Affiliation:
Mathematics Department, University of Minnesota, Minneapolis, MN 55455, USAam@math.umn.edu
V. MARKOVIC
Affiliation:
Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdomdbae@maths.warwick.ac.uk, markovic@maths.warwick.ac.uk
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Abstract

We make a detailed study of the relation of a euclidean convex region $\Omega \subset \mathbb C$ to $\mathrm{Dome} (\Omega)$. The dome is the relative boundary, in the upper halfspace model of hyperbolic space, of the hyperbolic convex hull of the complement of $\Omega$. The first result is to prove that the nearest point retraction $r: \Omega \to \mathrm{Dome} (\Omega)$ is 2-quasiconformal. The second is to establish precise estimates of the distortion of $r$ near $\partial \Omega$.

Keywords

30C7530F4030F4530F60
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 92 , Issue 3 , May 2006 , pp. 624 - 654
Copyright
2006 London Mathematical Society

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