About The Number Of Interior Point Of Finite Planar Point Set

In this paper we consider only finite planar point sets in which no three points are collinear. Let P be a finite planar point set with no three points collinear. Vertices of the convex hull of P are called vertices of P and the remaining points of P are called interior points of P. Let V(P) denote the set of vertices of P and I(P) = P1 denote the interior points of P. Let V(P1) denote the set of vertices of I(P) and I(P1) denote the interior points of P1. For any integer k ≥ 1 let g(k) be the smallest integer such that every set of points in the plane with no three collinear and at least g(k) interior points has a subset whose convex hull contains exactly k interior points. In this paper, we prove that any finite planar set P with |V(P)| = 3, |Pi| = 8, |V(P1)| = 8,7,5 contains a convex k ?gon (3≤k≤6) with exactly 3 interior points. Moreover we find a set P with 8 interior points which does not contain a k ?gon with exactly 3 interior points. Therefore we obtain g(3) > 8.