The hyperbolic metric and spherically convex regions (Q1612778)

It is known that convex regions \(\Omega\subset\mathbb{C}\) can be characterized in terms of the density \(\lambda_{\Omega}(w)\) of the hyperbolic metric \(\lambda_{\Omega}(w)|dw|\). In this paper, the authors derive analogous characterizations for spherically convex regions \(\Omega\) of the Riemann sphere in terms of the spherical density \((1+|w|^2)\lambda_{\Omega}(w)\) of the hyperbolic metric. The authors indicate how the characterizations of Euclidean convexity in the plane can be viewed as a limiting case of their results.