It is known that for every dimension $d\ge 2$ and every $k0$ such that for every $n$-point set $X\subset \mathbb R^d$ there exists a $k$-flat that intersects at least $c_{d,k} n^{d+1-k} - o(n^{d+1-k})$ of the $(d-k)$-dimensional simplices spanned by $X$. However, the optimal values of the constants $c_{d,k}$ are mostly unknown. The case $k=0$ (stabbing by a point) has received a great deal of attention. In this paper we focus on the case $k=1$ (stabbing by a line). Specifically, we try to determine the upper bounds yielded by two point sets, known as the "stretched grid" and the "stretched diagonal". Even though the calculations are independent of $n$, they are still very complicated, so we resort to analytical and numerical software methods. Surprisingly, for $d=4,5,6$ the stretched grid yields better bounds than the stretched diagonal (unlike for all cases $k=0$ and for the case $(d,k)=(3,1)$, in which both point sets yield the same bound). The stretched grid yields $c_{4,1}\leq 0.00457936$, $c_{5,1}\leq 0.000405335$, and $c_{6,1}\leq 0.0000291323$.

中文翻译：

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用一条线刺入单纯形的上限

已知对于每个维度 $d\ge 2$ 和每个 $k0$ 使得对于每一个 $n$-点集合 $X\subset\mathbb R^d$ 存在一个 $k$-flat 至少与 $c_{d,k} n^{d+1-k} 相交- o(n^{d+1-k})$ 的 $(dk)$ 维单纯形由 $X$ 跨越。然而，常量 $c_{d,k}$ 的最优值大多是未知的。$k=0$（被点刺）一案引起了广泛关注。在本文中，我们重点关注 $k=1$（被线刺伤）的情况。具体来说，我们试图确定由两个点集产生的上限，称为“拉伸网格”和“拉伸对角线”。尽管计算独立于$n$，但它们仍然非常复杂，因此我们求助于解析和数值软件方法。令人惊讶的是，对于 $d=4,5,6$，拉伸网格产生比拉伸对角线更好的边界（不同于所有情况 $k=0$ 和情况 $(d, k)=(3,1)$，其中两个点集产生相同的界限）。拉伸网格产生 $c_{4,1}\leq 0.00457936$、$c_{5,1}\leq 0.000405335$ 和 $c_{6,1}\leq 0.0000291323$。