线面排列包络的复杂性
The complexity of the envelope of line and plane arrangements
摘要：
超平面排列的一个小平面称为外部的，如果它恰好属于一个有界胞元。所有外部刻面的集合形成布置的包络。由维度$d$中的$n$超平面定义的简单布置的外部刻面的数量被假设为至少为$d{n-2 \choose d-1}$。在本文中，我们表明，对于4行或更多的简单安排，外部小平面的最小数目等于$2(n-1)$，对于5个平面或更多的简单安排，外部小平面的最小数目介于$\frac{n(n-2)+6}{3}$和$(n-4)(2n-3)+5$之间。
英文摘要：
A facet of an hyperplane arrangement is called external if it belongs to exactly one bounded cell. The set of all external facets forms the envelope of the arrangement. The number of external facets of a simple arrangement defined by $n$ hyperplanes in dimension $d$ is hypothesized to be at least $d{n-2 \choose d-1}$. In this note we show that, for simple arrangements of 4 lines or more, the minimum number of external facets is equal to $2(n-1)$, and for simple arrangements of 5 planes or more, the minimum number of external facets is between $\frac{n(n-2)+6}{3}$ and $(n-4)(2n-3)+5$.
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。
