Cells in any simple polygon formed by a planar point set

AbstractLet P be a finite point set in general position in the plane. We consider empty convex subsets of P such that the union of the subsets constitute a simple polygon S whose dual graph is a path, and every point in P is on the boundary of S. Denote the minimum number of the subsets in the simple polygons S's formed by P by fp(P), and define the maximum value of fp(P) by Fp(n) over all P with n points. We show that ⌈(4n-17)/15⌉⩽Fp(n)⩽⌊n/2⌋