Abstract
A planar set $P$ is said to be cover-decomposable if there is a constant $k=k(P)$ such that every $k$-fold covering of the plane with translates of $P$ can be decomposed into two coverings. It is known that open convex polygons are cover-decomposable. Here we show that closed, centrally symmetric convex polygons are also cover-decomposable. We also show that an infinite-fold covering of the plane with translates of $P$ can be decomposed into two infinite-fold coverings. Both results hold for coverings of any subset of the plane.
Highlights
A planar set P is said to be cover-decomposable if there is a constant k = k(P ) such that every k-fold covering of the plane with translates of P can be decomposed into two coverings
It is known that open convex polygons are cover-decomposable
The study of multiple coverings was initiated by Davenport and L
Summary
The study of multiple coverings was initiated by Davenport and L. Toth [10] proved that all open convex polygons are cover-decomposable. Toth [7] proved that (open and closed) concave quadrilaterals are not cover-decomposable Palvolgyi [9] refuted Pach’s conjecture He proved that open and closed sets with smooth boundary are not cover-decomposable. Matrai and Soukup [2] constructed an infinite-fold covering of the line by translates of a closed set, whose decomposability is independent of ZFC. We believe that it is not the case for coverings of the plane with translates of a convex closed set It follows directly from Theorem 2 that an infinite-fold covering of the plane with translates of a closed, convex, centrally symmetric polygon is decomposable into two coverings.
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