Document

# Convex Polygons in Cartesian Products

## File

LIPIcs.SoCG.2019.22.pdf
• Filesize: 0.61 MB
• 17 pages

## Acknowledgements

This work was initiated at the 2017 Fields Workshop on Discrete and Computational Geometry (Carleton University, Ottawa, ON, July 31 - August 4, 2017).

## Cite As

Jean-Lou De Carufel, Adrian Dumitrescu, Wouter Meulemans, Tim Ophelders, Claire Pennarun, Csaba D. Tóth, and Sander Verdonschot. Convex Polygons in Cartesian Products. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.SoCG.2019.22

## Abstract

We study several problems concerning convex polygons whose vertices lie in a Cartesian product of two sets of n real numbers (for short, grid). First, we prove that every such grid contains a convex polygon with Omega(log n) vertices and that this bound is tight up to a constant factor. We generalize this result to d dimensions (for a fixed d in N), and obtain a tight lower bound of Omega(log^{d-1}n) for the maximum number of points in convex position in a d-dimensional grid. Second, we present polynomial-time algorithms for computing the longest convex polygonal chain in a grid that contains no two points with the same x- or y-coordinate. We show that the maximum size of such a convex polygon can be efficiently approximated up to a factor of 2. Finally, we present exponential bounds on the maximum number of convex polygons in these grids, and for some restricted variants. These bounds are tight up to polynomial factors.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
##### Keywords
• Erdős-Szekeres theorem
• Cartesian product
• convexity
• polyhedron
• recursive construction
• approximation algorithm

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Imre Bárány. Random points and lattice points in convex bodies. Bulletin of the American Mathematical Society, 45:339-365, 2008.
2. Imre Bárány and János Pach. On the number of convex lattice polygons. Combinatorics, Probability and Computing, 1(4):295-302, 1992.
3. Imre Bárány and Anatoly Moiseevich Vershik. On the number of convex lattice polytopes. Geometric and Functional Analysis, 2(4):381-393, 1992.
4. Alexander Barvinok. Lattice points and lattice polytopes. In Jacob E. Goodman, Joseph O'Rourke, and Csaba D. Tóth, editors, Handbook of Discrete and Computational Geometry, chapter 7, pages 185-210. CRC Press, Boca Raton, FL, 3rd edition, 2017.
5. Boris Bukh and Jiří Matoušek. Erdős-Szekeres-type statements: Ramsey function and decidability in dimension 1. Manuscript, 2012. Available from: URL: https://arxiv.org/abs/1207.0705.
6. Jean Cardinal, Csaba D. Tóth, and David R. Wood. A note on independent hyperplanes and general position subsets in d-space. Journal of Geometry, 108(1):33-43, 2017.
7. Václav Chvátal and G. T. Klincsek. Finding largest convex subsets. Congressus Numerantium, 29:453-460, 1980.
8. David Conlon, Jacob Fox, János Pach, Benny Sudakov, and Andrew Suk. Ramsey-type results for semi-algebraic relations. Transactions of the American Mathematical Society, 366:5043-5065, 2014.
9. Adrian Dumitrescu, André Schulz, Adam Sheffer, and Csaba D. Tóth. Bounds on the maximum multiplicity of some common geometric graphs. SIAM Journal on Discrete Mathematics, 27(2):802-826, 2013.
10. Herbert Edelsbrunner and Leonidas J. Guibas. Topologically Sweeping an Arrangement. Journal of Computer and System Sciences, 38(1):165-194, 1989.
11. György Elekes, Melvyn B. Nathanson, and Imre Z. Ruzsa. Convexity and Sumsets. Journal of Number Theory, 83(2):194-201, 2000.
12. Paul Erdős. Appendix, in Klaus F. Roth, On a problem of Heilbronn. Journal of the London Mathematical Society, 26:198-204, 1951.
13. Paul Erdős. Some more problems on elementary geometry. Gazette of the Australian Mathematical Society, 5(2):52-54, 1978.
14. Paul Erdős and György Szekeres. On some extremum problems in elementary geometry. Annales Universitatis Scientiarium Budapestinensis de Rolando Eötvös Nominatae Sectio Mathematica, 3-4:53-62, 1960/1961.
15. Panos Giannopoulos, Christian Knauer, and Daniel Werner. On the Computational Complexity of Erdős-Szekeres and Related Problems in ℝ³. In Proc. 21st European Symposium on Algorithms, volume 8125 of LNCS, pages 541-552. Springer, 2013.
16. Ronald L. Graham. Euclidean Ramsey theory. In Jacob E. Goodman, Joseph O'Rourke, and Csaba D. Tóth, editors, Handbook of Discrete and Computational Geometry, chapter 11, pages 281-297. CRC Press, Boca Raton, FL, 3rd edition, 2017.
17. Richard K. Guy and Patrick A. Kelly. The No-Three-in-Line-Problem. Canadian Mathematical Bulletin, 11:527-531, 1968.
18. Norbert Hegyvári. On consecutive sums in sequences. Acta Mathematica Hungarica, 48(1-2):193-200, 1986.
19. Gyula Károlyi and Pavel Valtr. Configurations in d-space without large subsets in convex position. Discrete &Computational Geometry, 30(2):277-286, 2003.
20. Joseph S.B. Mitchell, Günter Rote, Gopalakrishnan Sundaram, and Gerhard Woeginger. Counting convex polygons in planar point sets. Information Processing Letters, 56(1):45-49, 1995.
21. Michael S. Payne and David R. Wood. On the general position subset selection problem. SIAM Journal on Discrete Mathematics, 27(4):1727-1733, 2013.
22. Attila Pór and David R. Wood. No-three-in-line-in-3D. Algorithmica, 47(4):481-488, 2007.
23. Orit E. Raz, Micha Sharir, and Ilya D. Shkredov. On the number of unit-area triangles spanned by convex grids in the plane. Computational Geometry: Theory and Applications, 62:25-33, 2017.
24. Orit E. Raz, Micha Sharir, and József Solymosi. Polynomials vanishing on grids: The Elekes-Rónyai problem revisited. American Journal of Mathematics, 138(4):1029-1065, 2016.
25. Orit E. Raz, Micha Sharir, and Frank De Zeeuw. Polynomials vanishing on Cartesian products: The Elekes-Szabó theorem revisited. Duke Mathematical Journal, 165(18):3517-3566, 2016.
26. David Alan Rosenthal. The classification of the order indiscernibles of real closed fields and other theories. PhD thesis, University of Wisconsin–Madison, 1981.
27. Tomasz Schoen and Ilya D. Shkredov. On Sumsets of Convex Sets. Combinatorics, Probability and Computing, 20(5):793-798, 2011.
28. Ryan Schwartz, József Solymosi, and Frank de Zeeuw. Extensions of a result of Elekes and Rónyai. Journal of Combinatorial Theory, Series A, 120(7):1695-1713, 2013.
29. Andrew Suk. On the Erdős-Szekeres convex polygon problem. Journal of the American Mathematical Society, 30:1047-1053, 2017.
30. Pavel Valtr. Sets in ℝ^d with no large empty convex subsets. Discrete Mathematics, 108(1-3):115-124, 1992.
31. Eric W. Weisstein. No-Three-in-a-Line-Problem. online, 2005. Available from: URL: http://mathworld.wolfram.com/No-Three-in-a-Line-Problem.html.
X

Feedback for Dagstuhl Publishing