  We completely characterize the boundedness of planar directional maximal
operators on L^p. More precisely, if Omega is a set of directions, we show that
M_Omega, the maximal operator associated to line segments in the directions
Omega, is unbounded on L^p, for all p < infinity, precisely when Omega admits
Kakeya-type sets. In fact, we show that if Omega does not admit Kakeya sets,
then Omega is a generalized lacunary set, and hence M_Omega is bounded on L^p,
for p>1.
