On Linear-Size Guillotine-Separable Subsets of Fat Convex Objects, Disks, and Squares

Abstract

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.