CLOSED SETS WITH THE KAKEYA PROPERTY

We say that a planar set A has the Kakeya property if there exist two different positions of A such that A can be continuously moved from the first position to the second within a set of arbitrarily small area. We prove that if A is closed and has the Kakeya property, then the union of the non-trivial connected components of A can be covered by a null set which is either the union of parallel lines or the union of concentric circles. In particular, if A is closed, connected and has the Kakeya property, then A can be covered by a line or a circle.