We prove that the bound on the Lp norms of the Kakeya type maximal functions studied by Cordoba [2] and Bourgain [1] are sharp for p>2. The proof is based on a construction originally due to Schoenberg [5], for which we provide an alternative derivation. We also show that r2 log (1/r) is the exact Minkowski dimension of the class of Kakeya sets in ℝ2, and prove that the exact Hausdorff dimension of these sets is between r2 log (1/r) and r2 log (1/r) [log log (1/r)]2+ε.