Restricting directions for Kakeya sets

We prove that the Kakeya maximal conjecture is equivalent to the \(\Omega\)-Kakeya maximal conjecture. This completes a recent result in [2] where Keleti and Math{é} proved that the Kakeya conjecture is equivalent to the \(\Omega\)-Kakeya conjecture. Moreover, we improve concrete bound on the Hausdorff dimension of a \(\Omega\)-Kakeya set : for any Bore set \(\Omega\) in S n--1 , we prove that if X \(\subset\) R n contains for any e \(\in\) \(\Omega\) a unit segment oriented along e then we have dX \(\ge\) 6 11 d\(\Omega\) + 1 where dE denotes the Hausdorff dimension of a set E.