论文标题
一条线的刺伤简单的上限
Upper bounds for stabbing simplices by a line
论文作者
论文摘要
众所周知，对于每个维度，$ d \ ge 2 $和每一个$ k <d $都存在一个常数$ c_ {d，k}> 0 $，以至于每个$ n $ point set $ x \ subset \ subset \ mathbb r^d $都存在$ k $ -flat，至少相交至少$ c_ {d，k} n^n^n^d^d^d+1-- o（n^{d+1-k}）$的$（d-k）$ - 尺寸简单由$ x $跨越。但是，常数的最佳值$ c_ {d，k} $大多未知。 CASE $ K = 0 $（点刺）受到了很大的关注。 在本文中，我们专注于$ k = 1 $的情况（一条线刺）。具体而言，我们尝试确定由两个点组所产生的上限，称为“拉伸网格”和“拉伸对角线”。即使计算独立于$ n $，它们仍然非常复杂，因此我们采用分析和数值软件方法。我们提供了强有力的证据表明，令人惊讶的是，对于$ d = 4,5,6 $，拉伸的网格产生的界限比伸展的对角线更好（与所有情况不同，$ k = 0 $，对于情况$（d，k）=（3,1）$，这两个点集均可产生相同的界限）。我们的实验表明，拉伸的网格产生$ c_ {4,1} \ leq 0.00457936 $，$ c_ {5,1} \ leq 0.000405335 $和$ c_ {6,1} \ leq 0.0000291323 $。
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 $c_{6,1}\leq 0.0000291323$.
