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.