Plane Finite Point Focus On An Empty Convex Polygon Count

Let P be a set of n points in the plane with no three collinear, that is, in general position. A region R is called empty if its interior contains no point of P, denoted by R ≌ . If for T P, CH(T) ≌ , the polygon determined by CH(T) is called empty convex polygon. We call a partition of P an empty convex partition if P is partitioned intosubsets S₁, S₂,......, S_t;= n, such that CH(S_i) is an empty convex |S_i|-gonfor each .Let k be a positive integer and N_k^∏(P) be the number of empty convex k-gons in an empty convex partition ∏ of P. We denotegk{P) =: maxN_k^∏(P) : ∏ is an empty convex partition of P} G_k(n) =: min{gk(P) : |P| = n}In this paper, we obtain the following results:G₄(n) , for n = 21 × 2⁽k-1 - 4(k ≥ 1).For k ≥ 3, let n(k, l) be the smallest integer such that any set of n(k, l) points in the plane, no three collinear, contains two different subsets Q₁ and Q₂, such that CH(Q₁) is an empty convex k-gon, CH(Q₂) is an empty convex l-gon, and CH(Q₁)∩CH(Q₂) = , where CH stands for the convex hull.Also, we give direct proofs of two famous results n(3,4) = 7[18] and n(4,5) ≤ 14[20].