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 \mathrm{conv , and k-hole if, in addition, no point of S \setminus X lies in \mathrm{conv . 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  ).
Given a set of points S \subseteq 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 Mari\'c (2019 ).
R
,
with
D
,
R
\mathrm{conv
of X
R
\mathrm{conv
the convex hull of X
A
). Another independent verification of this result was done by Mari\'c (2019