[Paper Review] Indecomposable coverings with unit discs.

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

Point processes and geometric inequalities5 references6 citations

TL;DR

This paper disproves Janos Pach's 1980 conjecture that every open convex set in the plane is cover-decomposable by proving that the unit disc is not cover-decomposable. The authors use a geometric construction involving m-fold coverings of the plane with unit discs to show that no finite m exists such that every m-fold covering can be decomposed into two coverings, thereby refuting the conjecture for smooth convex sets including the unit disc.

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is to disprove this conjecture by showing it does not hold for the unit disc (and thus refuting an argument in the unpublished manuscript [10] of Mani-Levitska and Pach from 1986, which the authors were kind enough to share with us).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.

Motivation & Objective

• To disprove Janos Pach's 1980 conjecture that every planar convex set is cover-decomposable.
• To demonstrate that the unit disc, despite being smooth and convex, cannot be decomposed from m-fold coverings into two coverings for any finite m.
• To generalize the result to any planar set with a smooth boundary, showing the failure of cover-decomposability in this class.
• To establish that the conjecture holds for unbounded sets, providing a contrast to the failure in the bounded, smooth case.
• To resolve an open problem posed by Peter Winkler in 2009 regarding the decomposability of 4-fold coverings with unit discs.

Proposed method

• Constructing an m-fold covering of the plane using translates of the unit disc that cannot be partitioned into two coverings.
• Using geometric and topological arguments to show that no finite m satisfies the cover-decomposability condition for the unit disc.
• Leveraging the smoothness of the boundary to generalize the counterexample beyond the unit disc to any smooth convex set.
• Applying techniques from hypergraph theory, particularly non-2-colorable hypergraphs realized by discs of varying sizes, to inform the construction.
• Using approximation by convex polygons to connect the result to discrete geometry and hypergraph coloring.
• Extending the result to higher dimensions by constructing a simpler counterexample than previously known.

Experimental results

Research questions

• RQ1 Is the unit disc cover-decomposable, meaning can every m-fold covering with unit discs be decomposed into two coverings for some finite m?
• RQ2 Does the cover-decomposability conjecture hold for all planar convex sets, including those with smooth boundaries?
• RQ3 Can the counterexample for the unit disc be generalized to other smooth convex sets?
• RQ4 Does the conjecture hold for unbounded convex sets, and if so, under what conditions?
• RQ5 Can the result be extended to higher-dimensional spaces, particularly for unit balls?

Key findings

• The unit disc is not cover-decomposable, thereby disproving Pach's 1980 conjecture for this specific case.
• The counterexample generalizes to any planar set with a smooth boundary, showing that such sets are not cover-decomposable.
• The result provides a simpler counterexample in higher dimensions than the one previously constructed in the unpublished manuscript [10].
• The construction demonstrates that even 4-fold coverings with unit discs cannot be decomposed into two coverings, contradicting a widely held belief.
• The failure of cover-decomposability for smooth convex sets contrasts with the validity of the conjecture for unbounded sets, which the paper proves to satisfy the variant condition.
• The work strengthens a result from [14] by constructing a non-2-colorable hypergraph realized by discs of varying sizes, now extended to the unit disc case.