# The Open Problems Project

Previous: Problem 65: Magic Configurations

# Problem 66: Reflexivity of Point Sets

Statement

Let \rho(S) be the fewest number of reflex vertices in a polygonization of a 2D point set S, i.e., the fewest reflexivities of any simple polygon whose vertex set is S. Let \rho(n) be the maximum of \rho(S) over all sets S with n points. What is \rho(n)?

Origin

[AFH+03]

Status/Conjectures

Open.

Partial and Related Results

In [AFH+03] the authors prove that \lfloor n/4 \rfloor \le \rho(n) \le \lceil n/2 \rceil and conjecture that \rho(n) = \lfloor n/4 \rfloor. The upper bound was recently improved to \frac{5}{12} n + O(1) \approx 0.4167 n in [AAK08].

Related Open Problems

Problem 16: Simple Polygonalizations.

Categories

polygons; point sets.

Entry Revision History

J. O’Rourke, 3 Aug. 2006; 16 Jul 2008.

# Bibliography

[AAK08]

Eyal Ackerman, Oswin Aichholzer, and Balazs Keszegh. Improved upper bounds on the reflexivity of point sets. Comput. Geom.: Theory Appl., 2008. To appear.

[AFH+03]

Esther M. Arkin, Sándor P. Fekete, Ferran Hurtado, Joseph S. B. Mitchell, Marc Noy, Vera Sacristán, and Saurabh Sethia. On the reflexivity of point sets. In B. Aronov, S. Basu, J. Pach, and M. Sharir, editors, Discrete and Computational Geometry: The Goodman-Pollack Festschrift, pages 139–156. Springer, 2003.