On the Existence of a Point Subset with 3 or 5 Interior

Xianglin Wei
Mathematics Department, Hebei Normal University, Shijiazhuang 050016,
P.R.China

Abstract    
An interior point of a finite planar point set is a point of the set that is not on the boundary of the convex hull of the set. Avis, Hosono, and Urabe discussed the existence of a point subset with 4 or 5 interior points. We obtain a result on the existence of a point subset with 3 or 5 interior points. For any integer k 1, let r(k) be the smallest integer such that every set of points in the plane, no three collinear, containing at least r(k) interior points has a subset of points containing k or k+2 interior point. We prove that r(3)=8.