Abstract
Let P be a set of n points in general position in the plane. The Second Selection Lemma states that for any family of \(\Theta(n^3)\) triangles spanned by P, there exists a point of the plane that lies in a constant fraction of them. For families of \(\Theta(n^{3-\alpha})\) triangles, with \(0\le \alpha \le 1\), there might not be a point in more than \(\Theta(n^{3-2\alpha})\) of those triangles. An empty triangle of P is a triangle spanned by P not containing any point of P in its interior. Bárány conjectured that there exists an edge spanned by P that is incident to a super-constant number of empty triangles of P. The number of empty triangles of P might be as low as \(\Theta(n^2)\); in such a case, on average, every edge spanned by P is incident to a constant number of empty triangles. The conjecture of Bárány suggests that for the class of empty triangles the above upper bound might not hold. In this paper we show that, somewhat surprisingly, the above upper bound does in fact hold for empty triangles. Specifically, we show that for any integer n and real number \(0\leq \alpha \leq 1\) there exists a point set of size n with \(\Theta(n^{3-\alpha})\) empty triangles such that any point of the plane is only in \(O(n^{3-2\alpha})\) empty triangles.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Alon, I. Bárány, Z. Füredi and D. J. Kleitman, Point selections and weak \(\epsilon\)-nets for convex hulls, Combin. Probab. Comput., 1 (1992), 189–200.
B. Aronov, B. Chazelle, H. Edelsbrunner, L. J. Guibas, M. Sharir and R. Wenger, Points and triangles in the plane and halving planes in space, Discrete Comput. Geom., 6 (1991), 435–442.
I. Bárány, A generalization of Carathéodory’s theorem, Discrete Math., 40 (1982), 141 – 152.
I. Bárány, Z. Füredi and L. Lovász, On the number of halving planes, Combinatorica, 10 (1990), 175–183.
I. Bárány and G. Károlyi, Problems and results around the Erdős-Szekeres convex polygon theorem, in: Discrete and Computational Geometry (Tokyo, 2000), Lecture Notes in Computer Science, vol. 2098, Springer (Berlin, 2001), pp. 91–105.
I. Bárány, J. Marckert and M. Reitzner, Many empty triangles have a common edge, Discrete Comput. Geom., 50 (2013), 244–252.
I. Bárány and P. Valtr, Planar point sets with a small number of empty convex polygons, Studia Sci. Math. Hungar., 41 (2004), 243–266.
E. Boros and Z. Füredi, The number of triangles covering the center of an n-set, Geom. Dedicata, 17 (1984), 69–77.
D. Eppstein, Improved bounds for intersecting triangles and halving planes, J. Combinatorial Theory Ser. A, 62 (1993), 176–182.
P. Erdős, On some unsolved problems in elementary geometry, Mat. Lapok (N.S.), 2 (1992), 1–10.
J. D. Horton, Sets with no empty convex 7-gons, Canad. Math. Bull., 26 (1983), 482–484.
M. Katchalski and A. Meir, On empty triangles determined by points in the plane, Acta Math. Hungar., 51 (1988), 323–328.
J. Matoušek, Lectures on Discrete Geometry, Graduate Texts in Math., vol. 212, Springer-Verlag (New York, 2002).
G. Nivasch and M. Sharir, Eppstein’s bound on intersecting triangles revisited, J. Combinatorial Theory Ser. A, 116 (2009), 494–497.
P. Valtr, Convex independent sets and 7-holes in restricted planar point sets, Discrete Comput. Geom., 7 (1992), 135–152.
P. Valtr, On the minimum number of empty polygons in planar point sets, Studia Sci. Math. Hungar., 30 (1995), 155–163.
Acknowledgement
Research on this work has been initiated at a workshop of the H2020-MSCA-RISE project 73499 - CONNECT, held in Barcelona in June 2017. We thank all participants for the good atmosphere as well as for discussions on the topic.
Author information
Authors and Affiliations
Corresponding author
Additional information
D.P. and B.V. were partially supported by the Austrian Science Fund within the collaborative DACH project Arrangements and Drawings as FWF project I 3340-N35.
This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 734922.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Fabila-Monroy, R., Hidalgo-Toscano, C., Perz, D. et al. No selection lemma for empty triangles. Acta Math. Hungar. 173, 52–73 (2024). https://doi.org/10.1007/s10474-024-01431-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-024-01431-0