Spherical convexity and hyperbolic metric

Abstract

Let $\Omega$ be a domain in $\mathbb{C}$ with hyperbolic metric ${\lambda }_{\Omega }(z)|dz|$ of Gaussian curvature $-4.$ Mejia and Minda proved in their 1990 paper that $\Omega$ is (Euclidean) convex if and only if $d(z,\partial \Omega ){\lambda }_{\Omega }(z)\ge 1/2$ for $z\in \Omega ,$ where $d(z,\partial \Omega )$ denotes the Euclidean distance from $z$ to the boundary $\partial \Omega .$ In the present note, we will provide similar characterizations of spherically convex domains in terms of the spherical density of the hyperbolic metric.

8 pages

References in corpus (1)

• Spherical density of hyperbolic metric and uniform perfectness