Fixing a hole

TL;DR

The paper proves that for any finite set in general position in Euclidean space, one can find arbitrarily large supersets also in general position that contain no large empty convex polygons, extending previous results.

Contribution

It generalizes earlier work by Horton and Valtr to higher dimensions, showing the existence of large hole-free supersets in general position.

Findings

Existence of arbitrarily large supersets with no large holes in general position.

Small perturbations of lattice points can eliminate large holes.

Extension of planar results to higher dimensions.

Abstract

We show that any finite $S \subset \mathbb{R}^d$ in general position has arbitrarily large supersets $T \supseteq S$ in general position with the property that $T$ contains no empty convex polygon, or hole, with $C_d$ points, where $C_d$ is an integer that depends only on the dimension $d$. This generalises results of Horton and Valtr which treat the case $S = \emptyset$. The key step in our proof, which may be of independent interest, is to show that there are arbitrarily small perturbations of the set of lattice points $[n]^d$ with no large holes.