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.KeywordsLower BoundConvex HullConvex SubsetInterior PointSmall IntegerThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.