The hyperbolic metric and spherically convex regions

Abstract

There are a number of characterizations of convex subregions $\Omega$ of the complex plane $\mathbb{C}$ in terms of the density $\lambda_{\Omega}(w)$ of the hyperbolic metric $\lambda_{\Omega}(w)|dw|$ for $\Omega$. We derive analogous characterizations for spherically convex regions $\Omega$ on the Riemann sphere $\mathbb{P}$ in terms of the spherical density $\mu_{\Omega}(w) = (1+|w|^{2})\lambda_{\Omega}(w)$ of the hyperbolic metric. A proper subregion $\Omega$ of $\mathbb{P}$ is spherically convex if for all pairs $A$, $B$ of points in $\Omega$ the spherical geodesic (the shorter arc of the great circle) joining $A$ and $B$ lies in $\Omega$. As a limiting case of our results we obtain known characterizations of convex regions in $\mathbb{C}$.