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- W2883802393 abstract "Given a set of points $S subseteq mathbb{R}^2$, a subset $X subseteq S$, $|X|=k$, is called $k$-gon if all points of $X$ lie on the boundary of the convex hull $mathrm{conv} (X)$, and $k$-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 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)." @default.
- W2883802393 created "2018-08-03" @default.
- W2883802393 creator A5078050536 @default.
- W2883802393 date "2018-07-27" @default.
- W2883802393 modified "2023-10-01" @default.
- W2883802393 title "On Disjoint Holes in Point Sets" @default.
- W2883802393 hasPublicationYear "2018" @default.
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