论文标题
网格的层数
The layer number of grids
论文作者
论文摘要
剥离过程定义如下：从有限点集开始$ x \ subset \ mathbb {r}^d $，我们反复删除当前一组点的凸赫尔的顶点。完全删除集合$ x $所需的剥离步骤的数量称为$ x $的层数。在本文中，我们研究了$ d $二维整数网格$ [n]^d $的层数。我们证明，对于每$ d \ geq 1 $，$ [n]^d $的层编号至少为$ω\ left（n^\ frac {2d} {d+1} \ right）$。另一方面，我们表明，对于每$ d \ geq 3 $，最多需要$ o（n^{d -9/11}）$步骤，以完全删除$ [n]^d $。我们的方法基于Har-Peled和Lidický使用的方法来解决二维情况的方法。
The peeling process is defined as follows: starting with a finite point set $X \subset \mathbb{R}^d$, we repeatedly remove the set of vertices of the convex hull of the current set of points. The number of peeling steps needed to completely delete the set $X$ is called the layer number of $X$. In this paper, we study the layer number of the $d$-dimensional integer grid $[n]^d$. We prove that for every $d \geq 1$, the layer number of $[n]^d$ is at least $Ω\left(n^\frac{2d}{d+1}\right)$. On the other hand, we show that for every $d\geq 3$, it takes at most $O(n^{d - 9/11})$ steps to fully remove $[n]^d$. Our approach is based on an enhancement of the method used by Har-Peled and Lidický for solving the 2-dimensional case.
