Stars of Empty Simplices

TL;DR

This paper studies the maximum number of empty simplices containing a given subset in a random point set, establishing asymptotic bounds and simplifying previous proofs for the degrees of such simplices.

Contribution

It provides new asymptotic bounds for the degrees of empty simplices in random point sets, improving and simplifying prior results.

Findings

${ m deg}_d(X)= heta(n)$ for random points in $K$

${ m deg}_1(X)= heta(n^{d-1})$ for random points in $K$

Simplified proof techniques for degree bounds

Abstract

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 ${\rm deg}_1(X)=\Theta(n^{d-1})$.