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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. 624-654. ISSN 0024-6115

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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 2-quasiconformal. 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:	0024-6115
Date:	May 2006
Volume:	Vol.92
Number:	No.3
Page Range:	pp. 624-654
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. 2. D. B. A. Epstein, A. Marden and V. Markovic, ‘Quasiconformal homeomorphisms and the convex hull boundary’, Ann. of Math. (2) 159 (2004) 305–336. 3. D. B. A. Epstein and V. Markovic, ‘The logarithmic spiral: a counter-example to the k = 2 conjecture’, Ann. of Math. (2) 161 (2005) 925–957. 4. R. Fehlman, ‘Über extremale quasikonforme abbildungen’, Comment. Math. Helv. 56 (1981) 558–580. 5. N. Lakic, ‘Substantial boundary points for plane domains and Gardiner’s conjecture’, Ann. Acad. Sci. Fenn. 25 (2000) 285–306. 6. Ch. Pommerenke, Boundary behaviour of conformal maps (Springer, Berlin, 1992). 7. E. Reich, ‘Extremal quasiconformal mappings of the disk’, Geometric function theory, Vol. 1, Handbook of complex analysis (ed. R. Kühnau; Elsevier, Amsterdam, 2001) 75–136.
URI:	http://wrap.warwick.ac.uk/id/eprint/696