AI 中文总结
研究\(d\)维整数网格\(\{1,\ldots,n\}^d\)的层数，通过分析\(P_i\)、\(Z_n\)及闵可夫斯基和，利用归一化格体积整数性与顶点估计，证明其层数为\(n^{2d/(d + 1)}\)阶，\(d\geq2\)时在所有非空子集上一致成立。
AI 中文摘要
有限点集的层数是通过反复移除其凸包顶点来删除该点集所需的迭代次数。安布鲁斯、许、彭和严猜想，对于每个固定的\(d\)，\(d\)维整数网格\(\{1,\ldots,n\}^d\)的层数为\(n^{2d/(d + 1)}\)阶。我们证明了这个猜想。设\(P_i\)是\(i\)步后剩余点集的凸包，\(Z_n\)是半径为\(n\)的欧几里得球内格点的凸包。对于每一步留下非空点集的情况，闵可夫斯基和\(P_{i + 1}+Z_n\)不包含\(P_i+Z_n\)的顶点。归一化格体积的整数性，连同\(Z_n\)的巴拉尼 - 拉曼顶点估计，给出了与\(i\)无关的体积减少下限。对\(i\)求和得到匹配的上限，即使\(P_i\)是低维的。对于\(d\geq2\)，相同的上限在\(\{1,\ldots,n\}^d\)的所有非空子集上一致成立。
英文摘要
The layer number of a finite point set is the number of iterations needed to delete it by repeatedly removing the vertices of its convex hull. Ambrus, Hsu, Peng, and Yan conjectured that the layer number of the $d$-dimensional integer grid $\{1,\ldots,n\}^d$ is of order $n^{2d/(d+1)}$ for every fixed $d$. We prove this conjecture. Let $P_i$ be the convex hull of the point set remaining after $i$ steps, and let $Z_n$ be the convex hull of the lattice points in the Euclidean ball of radius $n$. For every step that leaves a nonempty point set, the Minkowski sum $P_{i+1}+Z_n$ contains no vertex of $P_i+Z_n$. Integrality of normalized lattice volume, together with the Bárány--Larman vertex estimate for $Z_n$, gives a lower bound, independent of $i$, on the resulting volume decrease. Summing over $i$ yields the matching upper bound, even when $P_i$ is lower-dimensional. For $d\ge2$, the same upper bound holds uniformly over all nonempty subsets of $\{1,\ldots,n\}^d$.
