## Abstract

Given a set S of n points in the plane, the reflexivity of S, ρ(S), is the minimum number of reflex vertices in a simple polygonalization of S. Arkin et al. [E.M. Arkin, S.P. Fekete, F. Hurtado, J.S.B. Mitchell, M. Noy, V. Sacristán, S. Sethia, On the reflexivity of point sets, in: B. Aronov, S. Basu, J. Pach M. Sharir (Eds.), Discrete and Computational Geometry: The Goodman-Pollack Festschrift, Springer, 2003, pp. 139-156] proved that ρ(S)≤⌈n/2⌉ for any set S, and conjectured that the tight upper bound is ⌊n/4⌋. We show that the reflexivity of any set of n points is at most 37n+O(1)≈0.4286n. Using computer-aided abstract order type extension the upper bound can be further improved to 512n+O(1)≈0.4167n. We also present an algorithm to compute polygonalizations with at most this number of reflex vertices in O(nlogn) time.

Original language | English |
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Pages (from-to) | 241-249 |

Number of pages | 9 |

Journal | Computational Geometry: Theory and Applications |

Volume | 42 |

Issue number | 3 |

DOIs | |

State | Published - Apr 2009 |

Externally published | Yes |

### Bibliographical note

Funding Information:E-mail addresses: [email protected] (E. Ackerman), [email protected] (O. Aichholzer), [email protected] (B. Keszegh). 1 Supported by the Austrian FWF Joint Research Project ‘Industrial Geometry’ S9205-N12. 2 This work was done while visiting the School of Computing Science at Simon Fraser University.

## Keywords

- Polygonalization
- Reflexivity
- Steiner points

## ASJC Scopus subject areas

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