论文标题
$α$ - 均分布点集的层数
The layer number of $α$-evenly distributed point sets
论文作者
论文摘要
对于$ \ mathbb {r}^d $中的有限点设置，我们考虑了一个剥离过程，其中每个步骤都删除了凸面船体的顶点。给定点集的层$ l（x）$ $ x $定义为剥离过程的步骤数，以便在$ x $中删除所有点。众所周知，如果$ x $是$ \ mathbb {r}^d $中的一组随机点，那么$ l（x）$的期望是$θ（| x | x |^{2/（d+1）}）$，最近显示，如果$ x $是$ x $的$ x $，则是飞机上的Square Grid in Point of Plane Grid，则$ L（L（x）=>>>> => | x | |^|^| |^| |^| | |^| |^| | | | | | 在本文中，我们调查了$α$ - $ evenly分布点集的层数，价格为$α> 1 $；这些点集共享随机点集的规律性方面，但在更一般的设置中。一组晶格点也是$α$ - 分布点的设置，以$α> 1 $。我们发现，对于$ o（| x |^{3/4}）$的上限，对于$α$ - 分布点的层数，分布点设置$ x $在飞机上的单位磁盘中的$ x $，以$α> 1 $，并提供明显的结构，显示出显示该上限的增长率，无法提高。此外，我们给出$ O（| x |^{\ frac {d+1} {2d}}）$的上限，用于$α$ - evenly分布点的层编号，以$ \ m athbb {r}^d $ in单位球中的单位球中的$ x $，对于某些$α> 1 $和$ d $ d $ d $ d $ d $ d \ geq eq quq 3 $。
For a finite point set in $\mathbb{R}^d$, we consider a peeling process where the vertices of the convex hull are removed at each step. The layer number $L(X)$ of a given point set $X$ is defined as the number of steps of the peeling process in order to delete all points in $X$. It is known that if $X$ is a set of random points in $\mathbb{R}^d$, then the expectation of $L(X)$ is $Θ(|X|^{2/(d+1)})$, and recently it was shown that if $X$ is a point set of the square grid on the plane, then $L(X)=Θ(|X|^{2/3})$. In this paper, we investigate the layer number of $α$-evenly distributed point sets for $α>1$; these point sets share the regularity aspect of random point sets but in a more general setting. The set of lattice points is also an $α$-evenly distributed point set for some $α>1$. We find an upper bound of $O(|X|^{3/4})$ for the layer number of an $α$-evenly distributed point set $X$ in a unit disk on the plane for some $α>1$, and provide an explicit construction that shows the growth rate of this upper bound cannot be improved. In addition, we give an upper bound of $O(|X|^{\frac{d+1}{2d}})$ for the layer number of an $α$-evenly distributed point set $X$ in a unit ball in $\mathbb{R}^d$ for some $α>1$ and $d\geq 3$.
