  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.
