Plane Limited Spots For Empty Convex Polygons On The Border Issues

Let P be a set of n points in general position in the plane, and let T C P. CH(T) is called an empty polygon if no points of P lie in the interior of CH(T), where CH stands for the convex hull, and we simply say T is an empty polygon.A partition of P is called a convex partition if P is partitioned by k subsets S₁,S₂,...,S_k, such that each CH(S_i) is an |S_i|-gon. If no points of P lie in CH(S_i) for each i, we call this partition an empty partition of P.Let N^π{P) be the number of convex polygons in a partition π of P. We denote g(P) =: min{N^π(P) : π is an empty partition of P} G{n) =: max{g(P) ∶|P| = n}It has been shown that 「(n + 1)/4(?) ≤ G(n) ≤ 「9n/34(?) in [1]. In this paper, the upper bound of G(n) is improved.