Improved upper bounds on the reflexivity of point sets

Eyal Ackerman, Oswin Aichholzer, Balázs Keszegh

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
Pages (from-to)241-249
Number of pages9
JournalComputational Geometry: Theory and Applications
Issue number3
StatePublished - Apr 2009
Externally publishedYes

Bibliographical note

Funding Information:
E-mail addresses: (E. Ackerman), (O. Aichholzer), (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.


  • Polygonalization
  • Reflexivity
  • Steiner points

ASJC Scopus subject areas

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


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