Planar point sets with a small number of empty convex polygons, manuscript

A subset A of a finite set P of points in the plane is called an empty polygon, if each point of A is a vertex of the convex hull of A and the convex hull of A contains no other points of P. We construct a set of n points in general position in the plane with only ≈ 1.62n2 empty triangles, ≈ 1.94n2 empty quadrilaterals, ≈ 1.02n2 empty pentagons, and ≈ 0.2n2 empty hexagons. 1