Holes in 2-convex point sets

Oswin Aichholzer, Martin Balko, Thomas Hackl, Alexander Pilz, Pedro Ramos, Pavel Valtr, Birgit Vogtenhuber

Research output: Contribution to journalArticleResearchpeer-review

Abstract

Let S be a set of n points in the plane in general position (no three points from S are collinear). For a positive integer k, a k-hole in S is a convex polygon with k vertices from S and no points of S in its interior. For a positive integer l, a simple polygon P is l-convex if no straight line intersects the interior of P in more than l connected components. A point set S is l-convex if there exists an l-convex polygonization of S. Considering a typical Erdős–Szekeres-type problem, we show that every 2-convex point set of size n contains an Ω(log⁡n)-hole. In comparison, it is well known that there exist arbitrarily large point sets in general position with no 7-hole. Further, we show that our bound is tight by constructing 2-convex point sets in which every hole has size O(log⁡n).

Original languageEnglish
Pages (from-to)38-49
Number of pages12
JournalComputational Geometry: Theory and Applications
Volume74
DOIs
Publication statusPublished - 1 Oct 2018

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Point Sets
Convex Sets
Interior
Simple Polygon
Integer
Convex polygon
Collinear
Intersect
Connected Components
Straight Line
Large Set

ASJC Scopus subject areas

  • Computer Science Applications
  • Geometry and Topology
  • Control and Optimization
  • Computational Theory and Mathematics
  • Computational Mathematics

Cite this

Holes in 2-convex point sets. / Aichholzer, Oswin; Balko, Martin; Hackl, Thomas; Pilz, Alexander; Ramos, Pedro; Valtr, Pavel; Vogtenhuber, Birgit.

In: Computational Geometry: Theory and Applications, Vol. 74, 01.10.2018, p. 38-49.

Research output: Contribution to journalArticleResearchpeer-review

Aichholzer, Oswin ; Balko, Martin ; Hackl, Thomas ; Pilz, Alexander ; Ramos, Pedro ; Valtr, Pavel ; Vogtenhuber, Birgit. / Holes in 2-convex point sets. In: Computational Geometry: Theory and Applications. 2018 ; Vol. 74. pp. 38-49.
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