arXiv 2101.081922021-01-20math.CO

三维空间中的砖分割问题

Brick partition problems in three dimensions

Ilkyoo Choi, Minseong Kim, Kiwon Seo

备注：: 8 pages, 3 figures

摘要：

$d$维块是一个集合$I_1\times \cdots \times I_d$，其中每个$I_i$是一个区间.给定块$B$，$B$的块分区是将$B$划分为块。如果每条轴平行线至少与$\mathcal{P}_d$中的$k$块相交，则$d$维块的块分区$\mathcal{P}_d$是$k$穿透的。Bucic等人明确地询问了$d$维砖的$k$穿孔砖分区的最小尺寸$p(d, k)$。当$d=2$时，已知答案为$4(k-1)$。我们的第一个结果几乎决定了$p(3, k)$。也就是说，我们构造了一个$3$维砖的$k$穿透砖划分与$12k-15$部分，这是从已知的下限仅$1$。作为上述问题的推广，我们还寻求$d$维块的块分区$\mathcal{P}_d$的最小尺寸$s(d, k)$，其中每个轴平行平面至少与$\mathcal{P}_d$中的$k$块相交。我们通过确定所有$k$的$s(3, k)$来解决$3$维情况下的问题.

英文摘要：

A $d$-dimensional brick is a set $I_1\times \cdots \times I_d$ where each $I_i$ is an interval. Given a brick $B$, a brick partition of $B$ is a partition of $B$ into bricks. A brick partition $\mathcal{P}_d$ of a $d$-dimensional brick is $k$-piercing if every axis-parallel line intersects at least $k$ bricks in $\mathcal{P}_d$. Bucic et al. explicitly asked the minimum size $p(d, k)$ of a $k$-piercing brick partition of a $d$-dimensional brick. The answer is known to be $4(k-1)$ when $d=2$. Our first result almost determines $p(3, k)$. Namely, we construct a $k$-piercing brick partition of a $3$-dimensional brick with $12k-15$ parts, which is off by only $1$ from the known lower bound. As a generalization of the above question, we also seek the minimum size $s(d, k)$ of a brick partition $\mathcal{P}_d$ of a $d$-dimensional brick where each axis-parallel plane intersects at least $k$ bricks in $\mathcal{P}_d$. We resolve the question in the $3$-dimensional case by determining $s(3, k)$ for all $k$.

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。

摘要：

英文摘要：

相关论文