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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 ${rm deg}_1(X)=Theta(n^{d-1})$.
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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 motivation coming from the close relationship between empty simplices and terminal quotient singularities. They conjectured a classification of empty simplices of prime volume, modulo finitely many exceptions. Their conjecture was proved by Sankaran (1990) with a simplified proof by Bober (2009). The same classification was claimed by Barile et al. in 2011 for simplices of non-prime volume, but this statement was proved wrong by Blanco et al. (2016+). In this article we complete the classification of $4$-dimensional empty simplices. In doing so we correct and complete the classification claimed by Barile et al., and we also compute all the finitely many exceptions, by first proving an upper bound for their volume. The whole classification has: - One $3$-parameter family, consisting of simplices of width equal to one. - Two $2$-parameter families (the one in Mori et al., plus a second new one). - Forty-six $1$-parameter families (the 29 in Mori et al., plus 17 new ones). - $2461$ individual simplices not belonging to the above families, with volumes ranging between 29 and 419. We characterize the infinite families of empty simplices in terms of lower dimensional point configurations that they project to, with techniques that can be applied to higher dimensions and larger classes of lattice polytopes.
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 (White), based on the fact that they all have width one. But for dimension $4$ no complete classification is known. Haase and Ziegler (2000) enumerated all empty $4$-simplices up to determinant 1000 and based on their results conjectured that after determinant $179$ all empty $4$-simplices have width one or two. We prove this conjecture as follows: - We show that no empty $4$-simplex of width three or more can have determinant greater than 5058, by combining the recent classification of hollow 3-polytopes (Averkov, Krumpelmann and Weltge, 2017) with general methods from the geometry of numbers. - We continue the computations of Haase and Ziegler up to determinant 7600, and find that no new $4$-simplices of width larger than two arise. In particular, we give the whole list of empty $4$-simplices of width larger than two, which is as computed by Haase and Ziegler: There is a single empty $4$-simplex of width four (of determinant 101), and 178 empty $4$-simplices of width three, with determinants ranging from 41 to 179.
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 the volume of the random convex hull of $X_1,ldots,X_{r+1}$ are established in high dimensions, that is, as $r$ and $n$ tend to infinity simultaneously. This includes, Berry-Esseen-type central limit theorems, log-normal limit theorems, moderate and large deviations. Also different types of mod-$phi$ convergence are derived. The results heavily depend on the asymptotic growth of $r$ relative to $n$. For example, we prove that the fluctuations of the volume of the simplex are normal (respectively, log-normal) if $r=o(n)$ (respectively, $rsim alpha n$ for some $0 < alpha < 1$).
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 $|Y^n|_2$ of a random vector $Y^n$ distributed according to $mu^n$ satisfies a Gaussian thin-shell property: the distribution of $|Y^n|_2$ concentrates around a certain value $s_0$, and the fluctuations of $|Y^n|_2$ are approximately Gaussian with the order $1/sqrt{n}$. We apply this thin shell property to the study of rotationally invariant random simplices, simplices whose vertices consist of the origin as well as independent random vectors $Y_1^n,ldots,Y_p^n$ distributed according to $mu^n$. We show that the logarithmic volume of the resulting simplex exhibits highly Gaussian behavior, providing a generalizing and unifying setting for the objects considered in Grote-Kabluchko-Thale [Limit theorems for random simplices in high dimensions, ALEA, Lat. Am. J. Probab. Math. Stat. 16, 141--177 (2019)]. Finally, by relating the volumes of random simplices to random determinants, we show that if $A^n$ is an $n times n$ random matrix whose entries are independent standard Gaussian random variables, then there are explicit constants $c_0,c_1in(0,infty)$ and an absolute constant $Cin(0,infty)$ such that [sup_{ s in mathbb{R}} left| mathbb{P} left[ frac{ log mathrm{det}(A^n) - log(n-1)! - c_0 }{ sqrt{ frac{1}{2} log n + c_1 }} < s right] - int_{-infty}^s frac{e^{ - u^2/2} du}{ sqrt{ 2 pi }} right| < frac{C}{log^{3/2}n}, ] sharpening the $1/log^{1/3 + o(1)}n$ bound in Nguyen and Vu [Random matrices: Law of the determinant, Ann. Probab. 42 (1) (2014), 146--167].
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 in noncommutative dynamics. We characterize nc simplices in terms of their geometry and in terms of structural properties of their corresponding operator systems. There is a natural definition of nc Bauer simplex that generalizes the classical definition of a Bauer simplex. We show that a compact nc convex set is an nc Bauer simplex if and only if it is affinely homeomorphic to the nc state space of a unital C*-algebra, generalizing a classical result of Bauer for unital commutative C*-algebras. We obtain several applications to noncommutative dynamics. We show that the set of nc states of a C*-algebra that are invariant with respect to the action of a discrete group is an nc simplex. From this, we obtain a noncommutative ergodic decomposition theorem with uniqueness. Finally, we establish a new characterization of discrete groups with Kazhdans property (T) that extends a result of Glasner and Weiss. Specifically, we show that a discrete group has property (T) if and only if for every action of the group on a unital C*-algebra, the set of invariant states is affinely homeomorphic to the state space of a unital C*-algebra.
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