直线插入单形的上界
Upper bounds for stabbing simplices by a line
摘要：
已知对于每个维$d\ge 2$和每个$k<d$，存在一个常数$c_{d,k}>0$，使得对于每个$n$ -点集$X\subset \mathbb R^d$，存在一个$k$ -平坦，它至少与$X$所张成的$(d-k)$ -维单形的$c_{d,k} n^{d+1-k} - o(n^{d+1-k})$相交.然而，常数$c_{d,k}$的最佳值大多是未知的。$k=0$（被一个点刺伤）一案受到了极大的关注。在本文中，我们专注于$k=1$（刺伤线）的情况下。具体来说，我们试图确定两个点集产生的上界，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，因此我们采用解析和数值软件方法。我们提供了强有力的证据，令人惊讶的是，$d=4,5,6$拉伸网格产生更好的边界比拉伸对角线（不像所有情况下$k=0$和$(d,k)=(3,1)$，其中两个点集产生相同的边界）。我们的实验表明，拉伸网格产生$c_{4,1}\leq 0.00457936$，$c_{5,1}\leq 0.000405335$，和$c_{6,1}\leq 0.0000291323$。
英文摘要：
It is known that for every dimension $d\ge 2$ and every $k<d$ there exists a constant $c_{d,k}>0$ 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. We provide strong evidence that, 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). Our experiments indicate that the stretched grid yields $c_{4,1}\leq 0.00457936$, $c_{5,1}\leq 0.000405335$, and $c_{6,1}\leq 0.0000291323$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
