Few slopes without collinearity (Q1077713)

Consider \(n\) points in the plane such that no three of them are collinear. The regular \(n\)-gon shows that the connecting lines may determine as few as \(n\) slopes; the author shows here that only affine images of the regular \(n\)-gon have this property, applying earlier results after establishing that all points are extreme points of the convex hull of the point set. Denote by \(\alpha(n,e)\) the minimum number of slopes occurring in a set of \(n\) points, \(e\) of which are extreme points. Bounds for this number are given. For example, \[ \begin{aligned}\alpha(n,3)&=3n-6\qquad \text{(Theorem 4)},\\ \alpha(n,e)&\leq 2n+14\quad\text{ for all }n>e\geq 5 \qquad \text{(Theorem 6)},\\ n+2&\leq \alpha(n,n-1)\leq 2n-4\text{ for }n \text{ even}, 2n-2\text{ for }n \text{ odd } \text{(Theorem 7)}.\end{aligned} \]