Holes and Islands in Random Point Sets

Authors Martin Balko, Manfred Scheucher, Pavel Valtr

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Author Details

Martin Balko
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Manfred Scheucher
  • Institut für Mathematik, Technische Universität Berlin, Germany
Pavel Valtr
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
  • Department of Computer Science, ETH Zürich, Switzerland

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Martin Balko, Manfred Scheucher, and Pavel Valtr. Holes and Islands in Random Point Sets. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 14:1-14:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


For d ∈ ℕ, let S be a finite set of points in ℝ^d in general position. A set H of k points from S is a k-hole in S if all points from H lie on the boundary of the convex hull conv(H) of H and the interior of conv(H) does not contain any point from S. A set I of k points from S is a k-island in S if conv(I) ∩ S = I. Note that each k-hole in S is a k-island in S. For fixed positive integers d, k and a convex body K in ℝ^d with d-dimensional Lebesgue measure 1, let S be a set of n points chosen uniformly and independently at random from K. We show that the expected number of k-islands in S is in O(n^d). In the case k=d+1, we prove that the expected number of empty simplices (that is, (d+1)-holes) in S is at most 2^(d-1) ⋅ d! ⋅ binom(n,d). Our results improve and generalize previous bounds by Bárány and Füredi [I. Bárány and Z. Füredi, 1987], Valtr [P. Valtr, 1995], Fabila-Monroy and Huemer [Fabila-Monroy and Huemer, 2012], and Fabila-Monroy, Huemer, and Mitsche [Fabila-Monroy et al., 2015].

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatoric problems
  • Theory of computation → Computational geometry
  • stochastic geometry
  • random point set
  • Erdős-Szekeres type problem
  • k-hole
  • k-island
  • empty polytope
  • convex position
  • Horton set


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  1. O. Aichholzer, M. Balko, T. Hackl, J. Kynčl, I. Parada, M. Scheucher, P. Valtr, and B. Vogtenhuber. A Superlinear Lower Bound on the Number of 5-Holes. In 33rd International Symposium on Computational Geometry (SoCG 2017), volume 77 of Leibniz International Proceedings in Informatics, pages 8:1-8:16, 2017. Full version: https://arXiv.org/abs/1703.05253. URL: https://doi.org/10.4230/LIPIcs.SoCG.2017.8.
  2. G. E. Andrews, R. Askey, and R. Roy. Special functions, volume 71 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1999. URL: https://doi.org/10.1017/CBO9781107325937.
  3. J. Balogh, H. González-Aguilar, and G. Salazar. Large convex holes in random point sets. Computational Geometry, 46(6):725-733, 2013. URL: https://doi.org/10.1016/j.comgeo.2012.11.004.
  4. I. Bárány and Z. Füredi. Empty simplices in Euclidean space. Canadian Mathematical Bulletin, 30(4):436-445, 1987. URL: https://doi.org/10.4153/cmb-1987-064-1.
  5. I. Bárány and P. Valtr. Planar point sets with a small number of empty convex polygons. Studia Scientiarum Mathematicarum Hungarica, 41(2):243-266, 2004. URL: https://doi.org/10.1556/sscmath.41.2004.2.4.
  6. P. Erdős. Some more problems on elementary geometry. Australian Mathematical Society Gazette, 5:52-54, 1978. URL: http://www.renyi.hu/~p_erdos/1978-44.pdf.
  7. L. C. Evans and R. F. Gariepy. Measure theory and fine properties of functions. Textbooks in Mathematics. CRC Press, Boca Raton, FL, revised edition, 2015. Google Scholar
  8. R. Fabila-Monroy and C. Huemer. Covering Islands in Plane Point Sets. In Computational Geometry: XIV Spanish Meeting on Computational Geometry, EGC 2011, volume 7579 of Lecture Notes in Computer Science, pages 220-225. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-34191-5_21.
  9. R. Fabila-Monroy, C. Huemer, and D. Mitsche. Empty non-convex and convex four-gons in random point sets. Studia Scientiarum Mathematicarum Hungarica. A Quarterly of the Hungarian Academy of Sciences, 52(1):52-64, 2015. URL: https://doi.org/10.1556/SScMath.52.2015.1.1301.
  10. T. Gerken. Empty Convex Hexagons in Planar Point Sets. Discrete & Computational Geometry, 39(1):239-272, 2008. URL: https://doi.org/10.1007/s00454-007-9018-x.
  11. H. Harborth. Konvexe Fünfecke in ebenen Punktmengen. Elemente der Mathematik, 33:116-118, 1978. In German. URL: http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002079801.
  12. J. D. Horton. Sets with no empty convex 7-gons. Canadian Mathematical Bulletin, 26:482-484, 1983. URL: https://doi.org/10.4153/CMB-1983-077-8.
  13. M. Katchalski and A. Meir. On empty triangles determined by points in the plane. Acta Mathematica Hungarica, 51(3-4):323-328, 1988. URL: https://doi.org/10.1007/BF01903339.
  14. J. Matoušek. Lectures on discrete geometry, volume 212 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2002. URL: https://doi.org/10.1007/978-1-4613-0039-7.
  15. M. C. Nicolas. The Empty Hexagon Theorem. Discrete & Computational Geometry, 38(2):389-397, 2007. URL: https://doi.org/10.1007/s00454-007-1343-6.
  16. Matthias Reitzner and Daniel Temesvari. Stars of empty simplices, 2019. URL: http://arxiv.org/abs/1808.08734.
  17. M. Scheucher. Points, Lines, and Circles: Some Contributions to Combinatorial Geometry. PhD thesis, Technische Universität Berlin, Institut für Mathematik, 2019. Google Scholar
  18. P. Valtr. Sets in ℝ^d with no large empty convex subsets. Discrete Mathematics, 108(1):115-124, 1992. URL: https://doi.org/10.1016/0012-365X(92)90665-3.
  19. P. Valtr. On the minimum number of empty polygons in planar point sets. Studia Scientiarum Mathematicarum Hungarica, pages 155-163, 1995. URL: https://refubium.fu-berlin.de/handle/fub188/18741.
  20. P. Valtr. On empty hexagons. In Surveys on Discrete and Computational Geometry: Twenty Years Later, volume 453 of Contemporary Mathematics, pages 433-441. American Mathematical Society, 2008. URL: http://bookstore.ams.org/conm-453.
  21. P. Valtr. On empty pentagons and hexagons in planar point sets. In Proceedings of Computing: The Eighteenth Australasian Theory Symposium (CATS 2012), pages 47-48, Melbourne, Australia, 2012. URL: http://crpit.com/confpapers/CRPITV128Valtr.pdf.
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