Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Moments of the Maximal Number of Empty Simplices of a Random Point Set

Abstract

For a finite set X of n points from \( \mathbb {R}^M\), the degree of an M-element subset \(\{x_1,\dots ,x_M\}\) of X is defined as the number of M-simplices that can be constructed from this M-element subset using an additional point \(z \in X\), such that no further point of X lies in the interior of this M-simplex. Furthermore, the degree of X, denoted by \(\deg (X)\), is the maximal degree of any of its M-element subsets. The purpose of this paper is to show that the moments of the degree of X satisfy \(\mathbb {E}\,[ \deg (X)^k ] \ge c n^k/\log n\), for some constant \(c>0\), if the elements of the set X are chosen uniformly and independently from a convex body \(W \subset \mathbb {R}^M\). Additionally, it will be shown that \(\deg (X)\) converges in probability to infinity as the number of points of the set X goes to infinity.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. Oxford University Press, New York (2000)

  2. 2.

    Bárány, I., Károlyi, Gy.: Problems and results around the Erdős-Szekeres theorem. In: Akiyama, J., et al. (eds.) Discrete and Computational Geometry. Lecture Notes in Computer Science, vol. 2098, pp. 91–105. Springer, Berlin (2001).

  3. 3.

    Bárány, I., Marckert, J.-F., Reitzner, M.: Many empty triangles have a common edge. Discrete Comput. Geom. 50(1), 244–252 (2013)

  4. 4.

    Bárány, I., Valtr, P.: Planar point sets with a small number of empty convex polygons. Stud. Sci. Math. Hung. 41(2), 243–266 (2004)

  5. 5.

    Brass, P., Moser, W.O.J., Pach, J.: Research Problems in Discrete Geometry. Springer, New York (2005)

  6. 6.

    Erdős, P.: On some unsolved problems in elementary geometry. Mat. Lapok 2(2), 1–10 (1992) (in Hungarian).

  7. 7.

    Galerne, B.: Computation of the perimeter of measurable sets via their covariogram. Applications to random sets. Image Anal. Stereol. 30(1), 39–51 (2011)

  8. 8.

    Reitzner, M., Schulte, M., Thäle, C.: Limit theory for the Gilbert graph. Adv. Appl. Math. 88, 26–61 (2017)

  9. 9.

    Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and Its Applications, vol. 44. Cambridge University Press, Cambridge (1993)

  10. 10.

    Schneider, R., Weil, W.: Stochastic and Integral Geometry. Probability and Its Applications. Springer, Berlin (2008)

Download references

Acknowledgements

The author would like to thank Christoph Thäle and Julian Grote for helpful discussion concerning the topics of this paper. Furthermore, the author expresses his gratitude towards the referees for their suggestions regarding improvements of the paper.

Author information

Correspondence to Daniel Temesvari.

Additional information

Editor in Charge: János Pach

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Temesvari, D. Moments of the Maximal Number of Empty Simplices of a Random Point Set. Discrete Comput Geom 60, 646–664 (2018). https://doi.org/10.1007/s00454-018-9989-9

Download citation

Keywords

  • Random point set in \(\mathbb {R}^M\)
  • Empty simplex
  • Covariogram
  • Stochastic geometry

Mathematics Subject Classification

  • Primary 52A05
  • Secondary 52B05
  • 60D05