Dowker-type theorems for hyperconvex discs

A hyperconvex disc of radius r is a planar set with nonempty interior that is the intersection of closed circular discs of radius r. A convex disc-polygon of radius r is a set with nonempty interior that is the intersection of a finite number of closed circular discs of radius r. We prove that the maximum area and perimeter of convex disc-n-gons of radius r contained in a hyperconvex disc of radius r are concave functions of n, and the minimum area and perimeter of disc-n-gons of radius r containing a hyperconvex disc of radius r are convex functions of n. We also consider hyperbolic and spherical versions of these statements.