AI 中文总结
本研究探讨阻断一般位置点集张成的余维一单形所需最少点数，改进平面下界，给出偶数维矩曲线的下界及奇数维的最优构造，推进阻断猜想相关研究。
AI 中文摘要
我们研究$b_d(n)$，即与$\boldsymbol{\rm I\negmedspace R}^d$中处于一般位置的$n$点集张成的每个$(d-1)$-单形的相对内部相交所需的最少点数。在平面中，这是阻断猜想（Blocking Conjecture）中的参数。我们将已知的最佳一般平面下界改进为$b_2(n)\frac{41}{13}n-O(\frac{n}{\text{log }n})$。对于偶数维$2r$矩曲线上的$n$个点，我们证明至少需要$\frac{1}{r!}n^r\text{log }n-O_r(n^r)$个点才能刺穿其所有余维一单形的相对内部，这一数量比三角剖分中余维一面的数量多出一个$\text{log }n$因子。对于奇数维矩曲线上的等距点，我们构造了一个最优阻断集，其大小等于三角剖分中余维一面的最大数量。
英文摘要
We study $b_d(n)$, the minimum number of points needed to meet the relative interior of every $(d-1)$-simplex spanned by an $n$-point set in general position in $\mathbb{R}^d$. In the plane, this is the parameter from the Blocking Conjecture. We improve the best known general planar lower bound to $ b_2(n)\ge \frac{41}{13}n-O\left(\frac{n}{\log n}\right)$. For $n$ points on the moment curve in even dimension $2r$, we prove that at least $\frac{1}{r!}n^r\log n-O_r(n^r)$ points are needed to pierce the relative interior of all its codimension-one simplices, which exceeds the number of codimension-one faces in a triangulation by a $\log n$ factor. For equally spaced points on the moment curve in odd dimensions, we construct an optimal blocking set whose size equals the maximum number of codimension-one faces in a triangulation.
Comments12 pages, 1 figure
