given a connected region on a sphere, project it onto a plane. some shapes of regions necessarily have more distortion than others, no matter what map projection is used. ("amount of distortion" needs to be precisely defined.) which shapes?
it's not strictly area: a long thin band almost circling a great circle can be flattened into a rectangle without much distortion. or a star-shaped union of such bands. we only care about the internal distortion within a shape after projection, and not, say, that the distance between consecutive star endpoints is wrong.
perhaps the badness of a shape is proportional to the size of its largest inscribed circle (spherical cap).
consider only regions without internal holes. (merge the internal hole into the region if necessary.) map non-contiguous regions by treating each connected component separately (as done with Alaska and Hawaii on typical U.S. maps.)
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