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

AbstractLet P be a set of n points in the plane, no three collinear. A convex polygon of P is called empty if no point of P lies in its interior. An empty partition of P is a partition of P into empty convex polygons. Let k be a positive integer and Nkπ(P) be the number of empty convex k-gons in an empty partition π of P. Define gk(P)≕max{Nkπ(P):πis an empty partition of P}, Gk(n)≕min{gk(P):|P|=n}. We mainly study the case of k=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⌋