Multiple coverings with closed polygons

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.