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


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.

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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.

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Correspondence to Daniel Temesvari.

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Editor in Charge: János Pach

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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

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

Mathematics Subject Classification

  • Primary 52A05
  • Secondary 52B05
  • 60D05