A Note on the Value About a Disjoint Convex Partition Problem

Abstract

Let $P$ be a planar point set with no three points collinear. A $k$-hole of $P$ is a convex $k$-gon $H$ such that the vertices of $H$ are elements of $P$ and no element of $P$ lies inside $H$. In this article, we prove that for any planar $9$-point set $P$ with no three points collinear and at least $5$ vertices on the boundary of the convex hull, $P$ contains a $5$-hole and a disjoint $3$-hole.