Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 34, 2009, 447-473

QUASIHYPERBOLIC GEOMETRY OF PLANAR DOMAINS

Jussi Väisälä

Helsingin yliopisto, Matematiikan ja tilastotieteen laitos
PL 68, 00014 Helsinki, Finland; jussi.vaisala 'at' helsinki.fi

Abstract. Let k(a,b) denote the quasihyperbolic distance between points a, b in a domain G \subset R². We show that there is a universal constant c₀ > 0 with the following properties: (1) If k(a,b) < c₀, then there is only one quasihyperbolic geodesic from a to b. (2) If k(a,b) < c₀ and if \gamma is a quasihyperbolic geodesic from a to b, then there is a prolongation of \gamma to a quasihyperbolic geodesic \gamma₁ from a to b₁ with k(a,b₁) = c₀. (3) Each quasihyperbolic disk of radius r < c₀ is strictly convex in the euclidean metric.

2000 Mathematics Subject Classification: Primary 30C65.

Key words: Quasihyperbolic geodesic, quasihyperbolic ball, Voronoi diagram.

Reference to this article: J. Väisälä: Quasihyperbolic geometry of planar domains. Ann. Acad. Sci. Fenn. Math. 34 (2009), 447-473.

Full document as PDF file