Disjoint Partitions And Maximum Area Polygons In Planar Point Sets

Let P be a set of n points in the plane with no three collinear, that is, P is in general position. We call a partition of Fa disjoint partition if P is partitioned into subsets S₁, S₂, …, S_t =n, such that each CH(S_i) is an |S_i|-gon for every i = 1,2,…, t and CH(S_i) D CH(S_j)=(?) for any pair of indices i, j. Let k be a positive integer and ∏_k^Ï€(P) be the number of convex k-gons in a disjoint partition Ï€ of P. We denotef_k(P) =: max{∏_k^Ï€(P) : Ï€ is a disjoint partition of P}F_k(n) =: min{f_k(P) : |P| = n}In 2001 K.Hosono and M.Urabe considered the following question: How many disjoint empty convex k-gons can be constructed in a planar point set for a fixed k? They mainly studied case of k = 4 and posed some open problems. In this dissertation we discuss the problems about the values of F₄(26) and F₅(16) and make a progress toward the complete solution. Also, we give direct proofs of two famous results F₄(9) = 2 and F₅(10) = 1.A finite planar point set P is called to be in convex position if it forms the set of vertices of a convex polygon. Let P be in convex position. Denote the area of the convex hull of Q (?) P by S(Q). Let...