The Discrete Schwarz-Pick Lemma for Overlapping Circles

Jeff Van Eeuwen
1994 Proceedings of the American Mathematical Society  
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 n/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
more » ... 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.
doi:10.2307/2161218 fatcat:zntk3gm64fgrlk6rcpqql4xnrm