小型强力Epperimeter网
Small Strong Epsilon Nets
摘要：
设P是$\mathbb{R}^d$中的n个点的集合。点x被称为P的中心点，如果x包含在包含P的多于$dn\over d+1$点的每个凸对象中。我们称点x为对象族$\mathcal{C}$的强中心点，如果$x \in P$包含在包含P的多于常数分数的点的每个对象$C \in \mathcal{C}$中。强中心点甚至对于$\mathbb{R}^2$中的半空间也不存在。证明了$\mathbb{R}^d$中平行轴盒的强中心点的存在性，并给出了其精确界。然后，我们扩展到小的强$ε$ -网在平面上，并证明了上界和下界的$ε_i^\mathcal{S}$其中$\mathcal{S}$是家庭的轴平行矩形，半空间和磁盘。这里，$ε_i^\mathcal{S}$表示$[0,1]$中的最小实数，使得存在关于$\mathcal{S}$的大小为i的$ε_i^\mathcal{S}$ -网。
英文摘要：
Let P be a set of n points in $\mathbb{R}^d$. A point x is said to be a centerpoint of P if x is contained in every convex object that contains more than $dn\over d+1$ points of P. We call a point x a strong centerpoint for a family of objects $\mathcal{C}$ if $x \in P$ is contained in every object $C \in \mathcal{C}$ that contains more than a constant fraction of points of P. A strong centerpoint does not exist even for halfspaces in $\mathbb{R}^2$. We prove that a strong centerpoint exists for axis-parallel boxes in $\mathbb{R}^d$ and give exact bounds. We then extend this to small strong $ε$-nets in the plane and prove upper and lower bounds for $ε_i^\mathcal{S}$ where $\mathcal{S}$ is the family of axis-parallel rectangles, halfspaces and disks. Here $ε_i^\mathcal{S}$ represents the smallest real number in $[0,1]$ such that there exists an $ε_i^\mathcal{S}$-net of size i with respect to $\mathcal{S}$.
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。
