On disjoint holes in point sets

Abstract

Given a set of points $S \subseteq \mathbb{R}^2$, a subset $X \subseteq S$, $|X|=k$, is called \emph{$k$-gon} if all points of $X$ lie on the boundary of $\mathrm{conv} (X)$, and \emph{$k$-hole} if, in addition, no point of $S \setminus X$ lies in $\mathrm{conv} (X)$. We use computer assistance to show that every set of 17 points in general position admits two \emph{disjoint} 5-holes, that is, holes with disjoint respective convex hulls. This answers a question of Hosono and Urabe (2001).
In a recent article, Hosono and Urabe (2018) present new results on interior-disjoint holes -- a variant, which also has been investigated in the last two decades. Using our program, we show that every set of 15 points contains two interior-disjoint 5-holes. Moreover, our program can be used to verify that every set of 17 points contains a 6-gon within significantly smaller computation time than the original program by Szekeres and Peters (2006).