Suppose that we have a k-fold covering of the plane with translates of
a set S. Is it true that if k is large enough, then the covering can be
decomposed into two (or more) coverings? This simple question leads to
surprisingly hard problems, most of them still unsolved. Besides its theoretical
interest, the problem has important practical applications to sensor
networks. A planar set S is said to be cover-decomposable if there exists a
constant k = k(S) such that every k-fold covering of the plane with translates
of S can be decomposed into two coverings.
J. Pach proposed the problem of determining all cover-decomposable
sets, in 1980. According to his conjecture, all planar convex sets are coverdecomposable.
The conjecture has been veried only in some special cases,
in particular, it is known, that all open convex polygons are cover-decomposable.
All general positive results so far hold only for open sets.
Together with G. Toth we prove that closed, convex, centrally symmetric
polygons are also cover-decomposable. Cover-decoposability has many different
versions, we investigate the relationships between them and prove that
our result holds for each version. We also show that innite-fold coverings
can be decomposed into to innite-fold coverings.