On Disjoint Holes in Point Sets

07/27/2018

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by   Manfred Scheucher, et al.

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A k-hole in a point set S ⊆R^2 is a subset X ⊆ S, |X|=k, such that all points of X lie on the boundary of the convex hull conv (X) and no point of S ∖ 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.