A Note on Point Subsets with a Specified Number of Interior Points

Abstract

An interior point of a finite point set is a point of the set that is not on the boundary of the convex hull of the set. For any integer k ≥ 1, let g(k) be the smallest integer such that every set of points in the plane, no three collinear, containing at least g(k) interior points has a subset of points containing exactly k interior points. Similarly, for any integer k ≥ 3, let h(k) be the smallest integer such that every set of points in the plane, no three collinear, containing at least h(k) interior points has a subset of points containing exactly k or k+1 interior points. In this note, we show that g(k) ≥ 3k-1 for k ≥ 3. We also show that h(k) ≥ 2k+1 for 5 ≤ k ≤ 8, and h(k) ≥ 3k-7 for k ≥ 8.