发表机构
University of Copenhagen; Technion-IIT（哥本哈根大学; 以色列理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
我们证明了 Bezdek 关于木板覆盖稳定性的猜想：移除足够小的同心圆盘后，覆盖环带的木板族可重排覆盖整个圆盘，并给出适用于所有平面对称凸体的重叠下界估计。
AI 中文摘要
我们证明了 András Bezdek 关于平面圆盘 $D$ 的木板覆盖稳定性的猜想。具体而言，我们表明，如果从 $D$ 中移除一个足够小的同心圆盘 $rD$，那么覆盖所得环带的每一有限木板族都可以重新排列以覆盖整个圆盘。在证明中，我们展示了一个相关覆盖问题的强稳定性结果。如果洞的面积为 $a \geq 0$，则总重叠至少为 $c a/r$，其中 $c>0$ 是一个常数。该重叠估计并非圆盘特有，适用于所有平面对称凸体。我们开发了两个新工具：一个剪枝程序和一个展平机制。
英文摘要
We prove a longstanding conjecture of András Bezdek on the stability of plank covers of the planar disk $D$. If a sufficiently small disk $\varepsilon D$ is removed, then every finite family of planks covering the resulting annulus can be rearranged to cover the whole disk. For the proof, we establish a stronger dimension-independent stability theorem. If $K \subset {\mathbb R}^n$ is an origin-symmetric convex body for $n \geq 2$, and the uncovered region lies in $\varepsilon K$ and has volume $v$, then the excess overlap is at least $c v/\varepsilon$, where $c>0$ is an absolute constant. The proof introduces a pruning procedure and a flattening mechanism. Pruning peels away curvature through local removals whose cumulative effect is guaranteed to be substantial. Flattening trades straightness for bounded multiplicity and geometrically exposes the overlap that pays for each pruning step.
