Focus On Points In The Plane Does Not Cross The Convex Partition Further Discussion Of The Number Of Quadrilateral

We consider the problems about partitions of a planar set into disjoint convex polygons. In [1] K. Hosono and M. Urabe discussed the largest number of disjoint convex quadrilaterals in all disjoint partitions of a planar point set. Let P be a set of n points in the plane, no three collinear. Let k be a positive integer and πk (P) be the number of convex k-gons in a disjoint partition π of P. We denote by fk(P) the maximum value of πk (P) in all disjoint partitions π of P and we define Fk(n) = min{fk(P)} over all set P of n points. K. Hosono and M. Urabe showed that F4(n) > (3n- 1)/13 for an infinite series of specified values, that is, 0.2307n quadrilaterals are guaranteed. Therefore the outstanding question is whether F4(26) = 6 or not. We discuss this outstanding question and make an important progress toward the complete solution of the outstanding question.