AI 中文总结
研究任意大小的胖凸物体族是否有线性规模的可分离子集这一长期开放问题，通过证明每个两两不相交的胖凸物体族都有此类子集解决了该问题，结果扩展到高维，还改进了轴对齐正方形和圆盘可分离子集比例的界限。
AI 中文摘要
设\(\mathcal{K}\)是平面上一族两两不相交的物体。若子集\(\mathcal{K}^*\subseteq \mathcal{K}\)能通过一系列断头切割将\(\mathcal{K}^*\)中的所有物体彼此分离且不切割任何一个物体，则称其为“可分离的”。Urrutia（1996年）问任意\(n\)个凸物体的族是否有规模为\(\Omega(n)\)的可分离子集。Pach和Tardos（2000年）对线段给出否定回答，但对类似规模的胖物体给出肯定结果。最近，任意大小的轴对齐正方形集也有线性规模的可分离子集。然而，任意大小的胖凸物体集是否有线性规模的可分离子集仍是开放问题，对圆盘也如此。一个主要障碍是现有针对任意大小正方形的技术仅使用轴对齐切割，而即使对圆盘，仅轴对齐切割不足以得到线性规模的可分离子集。我们通过证明每个两两不相交的胖凸物体族都有线性规模的可分离子集解决了这个长期存在的开放问题。我们的结果扩展到更高维度：在\(\mathbb{R}^d\)（\(d\)为固定常数）中任意两两不相交、任意大小的胖凸物体族有线性规模的子集，可通过一系列超平面切割递归分离。我们的框架也为重要特殊情况给出改进保证。对于轴对齐正方形和轴对齐断头切割，利用正方形的额外结构性质表明至少\(13.46\%\)的正方形是可分离的，改进了Chalermsook、Kugelmann、Orgo、Uniyal和Zarsav（2025年）之前的最佳界限\(9/256 \approx 3.51\%\)。对于圆盘，利用Oler的填充不等式，证明至少\(n/93\)个圆盘总能被分离。
英文摘要
Let $\mathcal{K}$ be a family of pairwise disjoint objects in the plane. We say that a subset $\mathcal{K}^*\subseteq \mathcal{K}$ is \emph{separable} if it admits a sequence of guillotine cuts that separate all objects in $\mathcal{K}^*$ from each other while not cutting any of them. Urrutia (1996) asked whether any family of $n$ convex objects has a separable subset of size $Ω(n)$. Pach and Tardos (2000) answered this question negatively for line segments, but established positive results for fat objects of similar size. More recently, it was shown that sets of arbitrarily-sized axis-aligned squares also admit a separable subset of linear size. However, the question whether any set of arbitrarily-sized fat convex objects has a separable subset of linear size has remained open, even for disks. A major obstacle is that the existing technique for arbitrarily-sized squares uses only axis-aligned cuts, while even for disks, axis-aligned cuts alone are insufficient to obtain a separable subset of linear size. We resolve this longstanding open problem by proving that every family of pairwise disjoint fat convex objects has a separable subset of linear size. Our result extends to higher dimensions: any family of pairwise disjoint arbitrarily-sized fat convex objects in $\mathbb{R}^d$, where $d$ is a fixed constant, has a subset of linear size that is recursively separable by a sequence of hyperplane cuts. Our framework also yields improved guarantees for important special cases. For axis-aligned squares with axis-aligned guillotine cuts, we leverage additional structural properties of squares to show that at least $13.46\%$ of the squares are separable, improving the previous best bound of $9/256 \approx 3.51\%$ due to Chalermsook, Kugelmann, Orgo, Uniyal, and Zarsav (2025). For disks, by exploiting Oler's packing inequality, we prove that at least $n/93$ disks can always be separated.
