A k-hole in a point set S \subseteq R^2 is a subset X \subseteq S, |X|=k, such that all points of X lie on the boundary of the convex hull conv (X) and no point of S \setminus X lies in 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 from 2001. Moreover, we provide new bounds for three and more pairwise disjoint holes  .
Given a set of points S \subseteq 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 conv (X) , and k-hole if, in addition, no point of S \setminus X lies in 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 14 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) .
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A k-hole in a point set
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is called k-gon if
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from 2001. Moreover, we
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A
. 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 14 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)