# Spherical convexity and hyperbolic metric

## TL;DR

This paper characterizes spherically convex domains in the complex plane using the spherical density of the hyperbolic metric, extending previous Euclidean convexity results to spherical convexity.

## Contribution

It provides new criteria for spherically convex domains based on the spherical density of the hyperbolic metric, generalizing earlier Euclidean convexity characterizations.

## Key findings

- Characterization of spherically convex domains via spherical density.
- Extension of Euclidean convexity criteria to spherical convexity.
- New geometric conditions involving hyperbolic metric density.

## 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 $d(z,\partial\Omega)\lambda_\Omega(z)\ge1/2$ 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.