Plane Finite Set Of Empty Convex Partition Problem

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. Notice that we always regard T as empty polygons if |T| ≤ 2. 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 CH(S_i) ∩ CH(S_j) = φ, i ≠ j, then the partition is called a disjoint partition of P;If no points of P lie in CH(S_i) for each i, i.e., CH(S_i) ≌ Φ(P),we call this partition an empty partition of P, here CH(S_i) is allowed to intersect CH(Sj).Let k be a positive integer and N_k^π(P) be the number of convex k-gons in a partition π of P. Let N^π(P) be the number of convex polygons in a partition π of P. We denote :f_k(P) = max{N_k^π(P) : π is a disjiont partition of P},F_k(n) =: min{f_k(P) : |P| = n};f(P) =: min{N^π(P) : is is a disjiont partition of P},F(n) =: max{f(P) :|P| = n};g(P) =: mm{N^π(P) :π is an empty partition of P},G(n) =: max{g(P) : |P| = n}.These functions have been studied widely, and some important results have been obtained.In this paper, we define:g_k(P) =: max{N_k^π (P) : π is an empty partition of P}, G,(n) =: min{g_k(P) : |P| = n}.and obtain some important results about Gk(n): G4(13) = 3.G4(n) > LiJ-G4(n) > ^(n = 17 ? 2k¹ -4,k> 1).P be a set of 15 points, if |V(P)| = i (8 < i < 15), g5(P) > 2.Also, we give new and concise proofs of two famous results F(7) < 2 and G(ll) < 3.