Inscribable order types (Q6624174)

The orientation of an ordered triple of points in the plane is positive if the points appear in counterclockwise order around their convex hull, negative if they appear in clockwise order, and zero if they are collinear. A pair of finite point sets \(P, Q \subset \mathbb{R}^{2}\), which we call point configurations, have the same order type if there is a bijection that preserves the orientation of each triple, and this defines an equivalence relation. An order type \(\omega\) is an equivalence class of this relation, and we say that an element of \(\omega\) realises this \(\omega\). An order type is simple if there are no collinear triples. The extreme points of a configuration \(P\) are the vertices of the convex hull of \(P\), i.e. the points of \(P\) that can be isolated by a line, and the other points of \(P\) are the interior points. We say that a configuration is inscribed if its extreme points all lie on a circle, and we say that a configuration is inscribable if it has the same order type as an inscribed configuration, otherwise we say that it is uninscribable. In the paper under review, the authors provide (presumably) the first examples of uninscribable point configurations, and they provide the smallest one, called the non-Pascal configuration.\N\NThe authors then give the following several results which are devoted to inscribable configurations.\N\NTheorem 1. Every simple order type with at most two interior points is inscribable.\N\NTheorem 2. Every simple order type with at most five extreme points is inscribable.\N\NWe say that a point configuration \(P\) is minimally uninscribable if \(P\) is not inscribable, but \((P',B')\) is always inscribable, provided that \(P'\) is a proper subset of \(P\) and \(B' \subset P'\) is a subset of the extreme points of \(P\).\N\NTheorem 3. There are infinitely many minimally uninscribable order types.