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 $\textrm{deg} (X)$, 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 $\mathbb{E}\left[\textrm{deg} (X)^k\right] \geq c n^k / \log n$, 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 these moments converge in probability to infinity as the number of points of the set $X$ goes to infinity.