Stars of Empty Simplices

Let $X=\{x_1,\ldots,x_n\} \subset \mathbb R^d$ be an $n$-element point set in general position. For a $k$-element subset $\{x_{i_1},\ldots,x_{i_k}\} \subset X$ let the degree ${\rm deg}_k(x_{i_1},\ldots,x_{i_k})$ be the number of empty simplices $\{x_{i_1},\ldots,x_{i_{d+1}}\} \subset X$ containing no other point of $X$. The $k$-degree of the set $X$, denoted ${\rm deg}_k(X)$, is defined as the maximum degree over all $k$-element subset of $X$. We show that if $X$ is a random point set consisting of $n$ independently and uniformly chosen points from a compact set $K$ then ${\rm deg}_d(X)=\Theta(n)$, improving results previously obtained by B\'ar\'any, Marckert and Reitzner [Many empty triangles have a common edge, Discrete Comput. Geom., 2013] and Temesvari [Moments of the maximal number of empty simplices of a random point set, Discrete Comput. Geom., 2018] and giving the correct order of magnitude with a significantly simpler proof. Furthermore, we investigate ${\rm deg}_k(X)$. In the case $k=1$ we prove that ${\rm deg}_1(X)=\Theta(n^{d-1})$.