Inscribable Order Types

We call an order type inscribable if it is realized by a point configuration where all extreme points are all on a circle. In this paper, we investigate inscribability of order types. We first construct an infinite family of minimally uninscribable order types. The proof of uninscribability mainly uses Möbius transformations and the Frantz ellipse. We further show that every simple order type with at most two interior points is inscribable, and that the number of such order types is Θ (4^n/n^3/2) . We also suggest open problems around inscribability.