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

- 4 Downloads

## 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.

## Keywords

Random point set in \(\mathbb {R}^M\) Empty simplex Covariogram Stochastic geometry## Mathematics Subject Classification

Primary 52A05 Secondary 52B05 60D05## Notes

### 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.

## References

- 1.Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. Oxford University Press, New York (2000)MATHGoogle Scholar
- 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).Google Scholar
- 3.Bárány, I., Marckert, J.-F., Reitzner, M.: Many empty triangles have a common edge. Discrete Comput. Geom.
**50**(1), 244–252 (2013)MathSciNetCrossRefMATHGoogle Scholar - 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)MathSciNetMATHGoogle Scholar - 5.Brass, P., Moser, W.O.J., Pach, J.: Research Problems in Discrete Geometry. Springer, New York (2005)MATHGoogle Scholar
- 6.Erdős, P.: On some unsolved problems in elementary geometry. Mat. Lapok 2(2), 1–10 (1992) (in Hungarian).Google Scholar
- 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)MathSciNetCrossRefMATHGoogle Scholar - 8.Reitzner, M., Schulte, M., Thäle, C.: Limit theory for the Gilbert graph. Adv. Appl. Math.
**88**, 26–61 (2017)MathSciNetCrossRefMATHGoogle Scholar - 9.Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and Its Applications, vol. 44. Cambridge University Press, Cambridge (1993)CrossRefGoogle Scholar
- 10.Schneider, R., Weil, W.: Stochastic and Integral Geometry. Probability and Its Applications. Springer, Berlin (2008)CrossRefMATHGoogle Scholar