John disks, the Apollonian metric, and min-max properties

The main results of this paper are characterizations of John disks-the simply connected proper subdomains of the complex plane that satisfy a twisted double cone connectivity property. One of the characterizations of John disks is an analog of a result due to Gehring and Hag who found such a characterization for quasidisks. In both situations the geometric condition is an estimate for the domain's hyperbolic metric in terms of its Apollonian metric. The other characterization is in terms of an arc min-max property. © Indian Academy of Sciences.