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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 Barany, 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 ${rm deg}_1(X)=Theta(n^{d-1})$.
An empty simplex is a lattice simplex with only its vertices as lattice points. Their classification in dimension three was completed by White in 1964. In dimension four, the same task was started in 1988 by Mori, Morrison, and Morrison, with their m
A lattice $d$-simplex is the convex hull of $d+1$ affinely independent integer points in ${mathbb R}^d$. It is called empty if it contains no lattice point apart of its $d+1$ vertices. The classification of empty $3$-simplices is known since 1964 (Wh
Let $r=r(n)$ be a sequence of integers such that $rleq n$ and let $X_1,ldots,X_{r+1}$ be independent random points distributed according to the Gaussian, the Beta or the spherical distribution on $mathbb{R}^n$. Limit theorems for the log-volume and t
For fixed functions $G,H:[0,infty)to[0,infty)$, consider the rotationally invariant probability density on $mathbb{R}^n$ of the form [ mu^n(ds) = frac{1}{Z_n} G(|s|_2), e^{ - n H( |s|_2)} ds. ] We show that when $n$ is large, the Euclidean norm $|Y^n
We introduce a notion of noncommutative Choquet simplex, or briefly an nc simplex, that generalizes the classical notion of a simplex. While every simplex is an nc simplex, there are many more nc simplices. They arise naturally from C*-algebras and i