Abstract
Let P be a planar point set with no three points collinear; k points of P form a k-hole of P if these k points are the vertices of a convex polygon whose interior contains no points of P. Inthis article, we prove that any planar point set containing at least 13 points with no three points collinear contains pairwise disjoint 3-, 4-, and 5-holes if there exists a separating line SL4.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Erdős and G. Szekers, “A combinatorial problem in geometry,” Compositio Mathematica 2, 463–470 (1935).
P. Erdős and G. Szekers, “On some extremum problem in elementary geometry,” Ann. Univ. Sci. Budapest 3, 53–62 (1960).
G. Szekeres and L. Peters, “Computer solution to the 17-point Erdős-Szekeres problem,” ANZIAMJournal 48, 151–164 (2006).
H. Harborth, “Konvexe Funfeck in ebenen Punktmengen,” Element der Mathematic 33, 116–118 (1978).
J. Horton, “Sets with no empty convex 7-gons,” Canadian Mathematical Bulletin 26, 482–484 (1983).
T. Gerken, “Empty convex hexagons in planar point sets,” Discrete and Computational Geometry 39, 239–272 (2008).
C. Nicolás, “The empty hexagon theorem,” Discrete and Computational Geometry 38, 389–397 (2007).
K. Hosono and M. Urabe, “A minimal planar point set with specified disjoint empty convex subsets,” Lecture Notes in Computer Science 4535, 90–100 (2008).
M. Urabe, “On a partition into convex polygons,” Discrete AppliedMathematics 64, 179–191 (1996).
K. Hosono and M. Urabe, “On the number of disjoint convex quadrilaterals for a planar point set,” Computational Geometry: Theory and Applications 20, 97–104 (2001).
K. Hosono and M. Urabe, “On theminimum size of a point set containing two non-intersecting empty convex polygons,” Lecture Notes in Computer Science 3742, 117–122 (2005).
L. Wu and R. Ding, “Reconfirmation of two results on disjoint empty convex polygons,” Lecture Notes in Computer Science 4381, 216–220 (2007).
B. Bhattacharya and S. Das, “On the minimum size of a point set containing a 5-hole and a disjoint 4-hole,” Studia ScientiarumMathematicarum Hungarica 48, 445–457 (2011).
B. Bhattacharya and S. Das, Disjoint Empty Convex “Pentagons in Planar Point Sets,” Periodica Mathematica Hungarica 66, 73–86 (2013).
X. You and X. Wei, “On the minimum size of a point set containing a 5-hole and double disjoint 3-holes,” Mathematical Notes 97, 951–960 (2015).
X. You and X. Wei, “A note on the value about a disjoint convex partition problem,” Ars Combinatoria 115, 459–465 (2014).
X. You and X. Wei, “A note on the upper bound for disjoint convex partitions,” Mathematical Notes 96(1-2), 268–274 (2014).
Author information
Authors and Affiliations
Corresponding author
Additional information
The article was submitted by the authors for the English version of the journal. The corresponding author is Xinshang You.
Rights and permissions
About this article
Cite this article
You, X., Chen, T. A Note on the Value in the Disjoint Convex Partition Problem. Math Notes 104, 135–149 (2018). https://doi.org/10.1134/S0001434618070143
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434618070143