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arXiv · 2007.08972

On convex holes in $d$-dimensional point sets

Boris Bukh·Ting-Wei Chao·Ron Holzman
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Abstract

Given a finite set $A \subseteq \mathbb{R}^d$, points $a_1,a_2,\dotsc,a_{\ell} \in A$ form an $\ell$-hole in $A$ if they are the vertices of a convex polytope which contains no points of $A$ in its interior. We construct arbitrarily large point sets in general position in $\mathbb{R}^d$ having no holes of size $O(4^dd\log d)$ or more. This improves the previously known upper bound of order $d^{d+o(d)}$ due to Valtr. The basic version of our construction uses a certain type of equidistributed point sets, originating from numerical analysis, known as $(t,m,s)$-nets or $(t,s)$-sequences, yielding a bound of $2^{7d}$. The better bound is obtained using a variant of $(t,m,s)$-nets, obeying a relaxed equidistribution condition.

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Boris Bukh, Ting-Wei Chao, Ron Holzman. 2020-07-17. On convex holes in $d$-dimensional point sets. https://arxiv.org/abs/2007.08972

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