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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The discrete Schwarz-Pick lemma for overlapping circles


Author: Jeff Van Eeuwen
Journal: Proc. Amer. Math. Soc. 121 (1994), 1087-1091
MSC: Primary 30C80; Secondary 51M10, 52C15, 57M50
MathSciNet review: 1191873
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Abstract: Let P and $ P' $ be circle packings in the hyperbolic plane such that they are combinatorically equivalent, neighboring circles in P overlap one another at some fixed angle between 0 and $ \pi /2$ and the corresponding pairs of circles in $ P' $ overlap at the same angle, and the radius for any boundary circle of P is less than or equal to that of the corresponding boundary circle of $ P' $. In this paper we show that the radius of any interior circle of P is less than or equal to that of the corresponding circle in $ P' $, and the hyperbolic distance between the centers of circles in P is less than or equal to the distance between the corresponding circles in $ P' $. Furthermore, a single instance of finite equality in either of the above implies equality for all.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1191873-3
PII: S 0002-9939(1994)1191873-3
Article copyright: © Copyright 1994 American Mathematical Society