On the number of empty convex quadrilaterals of a finite set in the plane

Let PP be a set of nn points in the plane, no three collinear. A convex polygon of PP is called empty if no point of PP lies in its interior. An empty partition of PP is a partition of PP into empty convex polygons. Let kk be a positive integer and Nkπ(P) be the number of empty convex kk-gons in an empty partition ππ of PP. Define gk(P)≕max{Nkπ(P):πis an empty partition of P}, Gk(n)≕min{gk(P):|P|=n}Gk(n)≕min{gk(P):|P|=n}. We mainly study the case of k=4k=4 and get the result that G4(n)≥⌊9n38⌋. For specified n=21×2k−1−4(k≥1), we obtain the better bound G4(n)≥⌊5n−121⌋.