2021/08/16 by David Conlon, Conlon, David, Jeck Lim +1 · 1 citation
Computer Science · Mathematics · #52C07 #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #FOS: Mathematics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2108.07087
openalex publication_date 2021/08/16 · openalex created_date 2022/08/31 · openalex updated_date 2026/07/28
We show that any finite S ⊂ ℝd in general position has arbitrarily large supersets T ⊇ S in general position with the property that T contains no empty convex polygon, or hole, with Cd points, where Cd is an integer that depends only on the dimension d. This generalises results of Horton and Valtr which treat the case S = ∅. 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.