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Convex regions in the plane and their domes
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Epstein, D. B. A., Marden, A. and Markovic, V. (Vladimir). (2006) Convex regions in the plane and their domes. Proceedings of the London Mathematical Society, Vol.92 (No.3). pp. 624654. ISSN 00246115

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Official URL: http://dx.doi.org/10.1017/S002461150501573X
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 2quasiconformal. The second is to establish precise estimates of the distortion of $r$ near $\partial \Omega$.
Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Mathematics 
Library of Congress Subject Headings (LCSH):  Convex domains, Convex geometry, Geometry, Hyperbolic, Geometry, Plane, Algebraic number theory 
Journal or Publication Title:  Proceedings of the London Mathematical Society 
Publisher:  Cambridge University Press 
ISSN:  00246115 
Official Date:  May 2006 
Volume:  Vol.92 
Number:  No.3 
Page Range:  pp. 624654 
Identification Number:  10.1017/S002461150501573X 
Status:  Peer Reviewed 
Access rights to Published version:  Open Access 
References:  1. D. B. A. Epstein and A. Marden, ‘Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces’, Analytical and geometric aspects of hyperbolic space (ed. D. B. A. Epstein), London Mathematical Society Lecture Note Series 111 (Cambridge University Press, 1987) 113–253. 
URI:  http://wrap.warwick.ac.uk/id/eprint/696 
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