Stars of Empty Simplices

Let $X=\{x_1,\dots ,x_n\}\subset {\mathbb{R}}^d$ be an $n$-element point set in general position. For a $k$-element subset ${x_{i_1},\ldots,x_{i_k}} \subset X$letthedegree${\rm deg}k(x{i_1},\ldots,x_{i_k})$ be the number of empty simplices $\{x_{i_1},\dots ,x_{i_{d+1}}\}\subset X$ containing no other point of $X$. The $k$-degree of the set $X$, denoted ${\mathrm{d}\mathrm{e}\mathrm{g}}_k(X)$, 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 ${\mathrm{d}\mathrm{e}\mathrm{g}}_d(X)=\mathrm{\Theta }(n)$, 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 ${\rm deg}_k(X)$\mathrm{.}Inthecase$k=1$weprovethat${\rm deg}_1(X)=\Theta(n^{d-1})$.