[论文解读] Indecomposable coverings with unit discs.

Dömötör Pálvölgyi|arXiv (Cornell University)|Oct 25, 2013

Point processes and geometric inequalities参考文献 5被引用 6

一句话总结

本文通过证明单位圆盘不可覆盖分解，推翻了雅诺什·帕奇1980年提出的猜想，即平面上每个开凸集都是可覆盖分解的。作者通过使用单位圆盘对平面进行m重覆盖的几何构造，证明不存在有限的m使得每个m重覆盖都能分解为两个覆盖，从而否定了光滑凸集（包括单位圆盘）的该猜想。

ABSTRACT

We disprove the 1980 conjecture of Janos Pach about the cover-decomposability of open convex sets by showing that the unit disc is not cover-decomposable. In fact, our proof easily generalizes to any set with a smooth boundary. We also show that (the suitable variant of) the conjecture holds for unbounded sets. Let C be a collection of sets in R2. We say that C is an m-fold covering if every point of R2 is contained in at least m members of C. A 1-fold covering is simply called a covering. De nition. A planar set C is said to be cover-decomposable if there exists a (minimal) constant m = m(C) such that every m-fold covering of the plane with translates of C can be decomposed into two coverings. The problem of characterizing all cover-decomposable sets in the plane was proposed by Pach [12] in 1980. He made the following conjecture. Conjecture (Pach). Every planar convex set C is cover-decomposable. The goal of this paper is to disprove this conjecture by showing it does not hold for the unit disc (and thus refuting an argument in the unpublished manuscript [10] of Mani-Levitska and Pach from 1986, which the authors were kind enough to share with us). Theorem 1. The unit disc is not cover-decomposable. It is quite surprising that the conjecture fails already for discs. This special case of the problem was also mentioned as an open problem in the November 2009 issue of the Communications of the ACM by Peter Winkler [18], and he guessed (like everybody else as well) that it should be possible to decompose every 4-fold covering of the plane with unit discs into two coverings. For the proof, see Section 1 and 2. As a consequence, we also obtain a construction for the respective claim in higher dimensions about unit balls, giving a simpler example than the one in [10]. Theorem 1 is also a strengthening of a theorem of [14], where a non-2-colorable hypergraph is realized by discs (of various size). Since unit discs can be approximated by (convex) polygons, we can also derive †Institute of Mathematics, Eotvos University, Budapest Research supported by Hungarian National Science Fund (OTKA), grant PD 104386, the EUROGIGA project 10-EuroGIGA-OP-003 (OTKA NN 102029) and the Janos Bolyai Research Scholarship of the Hungarian Academy of Sciences.

研究动机与目标

• 推翻雅诺什·帕奇1980年提出的猜想，即平面上每个凸集都是可覆盖分解的。
• 证明单位圆盘尽管光滑且凸，但对任意有限的m，其m重覆盖都无法分解为两个覆盖。
• 将结果推广至任意具有光滑边界的一般平面集，表明该类集合不具备覆盖分解性。
• 证明该猜想对无界集成立，从而与有界光滑情形的失败形成对比。
• 解决彼得·温克勒2009年提出的开放问题，即单位圆盘的4重覆盖是否可分解。

提出的方法

• 通过平移单位圆盘构造一个无法划分为两个覆盖的m重覆盖。
• 运用几何与拓扑论证，证明对单位圆盘而言，不存在满足覆盖分解条件的有限m。
• 利用边界光滑性，将反例推广至任意光滑凸集。
• 应用超图理论技术，特别是利用不同大小圆盘实现的非2-可着色超图，以指导构造过程。
• 通过凸多边形逼近，将结果与离散几何及超图着色联系起来。
• 通过构造比先前已知更简单的反例，将结果推广至高维空间。

实验结果

研究问题

• RQ1 单位圆盘是否可覆盖分解，即是否存在某个有限的m，使得每个单位圆盘的m重覆盖均可分解为两个覆盖？
• RQ2 该覆盖分解猜想是否对所有平面凸集（包括具有光滑边界者）均成立？
• RQ3 单位圆盘的反例能否推广至其他光滑凸集？
• RQ4 该猜想是否对无界凸集成立？若成立，其条件为何？
• RQ5 该结果能否推广至高维空间，特别是单位球的情形？

主要发现

• 单位圆盘不可覆盖分解，从而否定了帕奇1980年猜想在该特定情况下的成立。
• 该反例可推广至任意具有光滑边界的平面集，表明此类集合不具备覆盖分解性。
• 该结果在高维空间中提供了比先前未发表手稿[10]中构造更简单的反例。
• 该构造表明，甚至4重覆盖的单位圆盘也无法分解为两个覆盖，与广泛持有的信念相矛盾。
• 光滑凸集的覆盖分解失败，与无界集情形下猜想成立形成鲜明对比，本文证明了后者满足该变体条件。
• 本工作通过构造由不同大小圆盘实现的非2-可着色超图，强化了[14]中的结果，现该结果已扩展至单位圆盘情形。