The "order type" of a finite point set was introduced about 40 years ago by Eli Goodman and Ricky Pollack. In this talk, I will discuss a new result on this notion obtained in a joint work with Emo Welzl: we show that the average number of extreme points in an $n$-point order type, chosen uniformly at random from all such order types, is $4+o(1)$.
Our proof hinges on the study of the projective order type of the antipodal $2n$-point set of the sphere, and more specifically of their orientation preserving bijections. Along the way, we show that these "symmetry groups" can be characterized along the lines of Felix Klein's analysis of the finite groups of rotations in 3-space.
Extended abstract from SoCG 2020: https://drops.dagstuhl.de/opus/volltexte/2020/12207/