  We consider the maximal operator with respect to uncentered cubes on
Euclidean space with arbitrary dimension. We prove that for any function with
bounded variation, the variation of its maximal function is bounded by the
variation of the function times a dimensional constant.
  We also prove the corresponding result for maximal operators with respect to
collections of more general sets than cubes. The sets are required to satisfy a
certain inner cone star condition and in addition the collection must enjoy a
tiling property which for example the collection of all cubes does enjoy and
the collection of all Euclidean balls does not.
