论文标题
凸层编号的新估计
New estimates for convex layer numbers
论文作者
论文摘要
从有限点集合开始，$ x \ subset \ mathbf {r}^d $，剥离过程反复删除当前集合的凸壳的顶点集。完全删除$ x $所需的剥离步骤的数量称为$ x $的层数，由$ l（x）$表示。在文章中，我们研究了$ b^d $，$ d $维单位球中包含的点集均匀分布的均匀分布家族的层数。这些集合由$ b^d $中的点组成，其最小距离尽可能大。我们表明，对于属于一个均匀分布的家族的集合$ x $，$ l（x）\ geqω（| x |^{1/d}）$持有，界限渐近均匀。另一方面，基于较早的结果，我们证明了$ l（x）\ leq o（| x |^{2/d}）$对$ d \ geq 2 $持有，它在$ o（| x |^{（d+1）/2d}的当前上限上，$ d \ deq geq 3 $都大大改善。最后，我们提供了均匀分布的家庭的递归结构，其集合满足$ l（x）=θ（| x |^{2/d -1/（d 2^{d -1}）}）$，表明我们的上限几乎很紧。
Starting with a finite point set $X \subset \mathbf{R}^d$, the peeling process repeatedly removes the set of the vertices of the convex hull of the current set. The number of peeling steps required to completely remove $X$ is called the layer number of $X$, denoted by $L(X)$. In the article, we study the layer number of evenly distributed families of point sets contained in $B^d$, the $d$-dimensional unit ball. These sets consist of points in $B^d$ whose minimal distance is asymptotically as large as possible. We show that for a set $X$ belonging to an evenly distributed family, $L(X) \geq Ω(|X|^{1/d})$ holds, with the bound being asymptotically sharp. On the other hand, building on earlier results, we prove that $L(X)\leq O(|X|^{2/d})$ holds for $d\geq 2$, which improves greatly on the current upper bound of $O(|X|^{(d+1)/2d})$ for $d \geq 3$. Finally, we provide a recursive construction of evenly distributed families whose sets satisfy $L(X) = Θ(|X|^{2/d - 1/(d 2^{d-1})})$, showing that our upper bound is nearly tight.
