Two Disjoint 5-Holes in Point Sets

Given a set of points $S \subseteq \mathbb{R}^2$, a subset $X \subseteq S$ with $|X|=k$ is called $k$-gon if all points of $X$ lie on the boundary of the convex hull of $X$, and $k$-hole if, in addition, no point of $S \setminus X$ lies in the convex hull of $X$. We use computer assistance to show that every set of 17 points in general position admits two disjoint 5-holes, that is, holes with disjoint respective convex hulls. This answers a question of Hosono and Urabe (2001). We also provide new bounds for three and more pairwise disjoint holes. 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). Another independent verification of this result was done by Marić (2019).

No reviews yet.

No comments yet.