### 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(logn)-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(logn).

Original language | English |
---|---|

Pages (from-to) | 38-49 |

Number of pages | 12 |

Journal | Computational Geometry: Theory and Applications |

Volume | 74 |

DOIs | |

Publication status | Published - 1 Oct 2018 |

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### ASJC Scopus subject areas

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

### Cite this

*Computational Geometry: Theory and Applications*,

*74*, 38-49. https://doi.org/10.1016/j.comgeo.2018.06.002

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

Research output: Contribution to journal › Article › Research › peer-review

*Computational Geometry: Theory and Applications*, vol. 74, pp. 38-49. https://doi.org/10.1016/j.comgeo.2018.06.002

}

TY - JOUR

T1 - Holes in 2-convex point sets

AU - Aichholzer, Oswin

AU - Balko, Martin

AU - Hackl, Thomas

AU - Pilz, Alexander

AU - Ramos, Pedro

AU - Valtr, Pavel

AU - Vogtenhuber, Birgit

PY - 2018/10/1

Y1 - 2018/10/1

N2 - 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(logn)-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(logn).

AB - 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(logn)-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(logn).

UR - http://www.scopus.com/inward/record.url?scp=85048744397&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2018.06.002

DO - 10.1016/j.comgeo.2018.06.002

M3 - Article

VL - 74

SP - 38

EP - 49

JO - Computational geometry

JF - Computational geometry

SN - 0925-7721

ER -