Geodesic
DOMAINS WITH CONVEX HYPERBOLIC RADIUS
Acta mathematica Universitatis Comenianae, Tome 70 (2001) no. 2 
L. V. Kovalev. DOMAINS WITH CONVEX HYPERBOLIC RADIUS. Acta mathematica Universitatis Comenianae, Tome 70 (2001) no. 2. http://geodesic.mathdoc.fr/item/AMUC_2001_70_2_a5/
@article{AMUC_2001_70_2_a5,
     author = {L. V. Kovalev},
     title = {DOMAINS {WITH} {CONVEX} {HYPERBOLIC} {RADIUS}},
     journal = {Acta mathematica Universitatis Comenianae},
     year = {2001},
     volume = {70},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/AMUC_2001_70_2_a5/}
}
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Voir la notice de l'article provenant de la source Comenius University

The hyperbolic radius of a domain on the Riemann sphere is equal to the reciprocal of the density of the hyperbolic metric. In the present paper, it is proved that the hyperbolic radius is a convex function if and only if the complement of the domain is a convex set.