In this fascinating paper the author establishes improved \(L^p\to L^q\) bounds for Kakeya and Besicovich type maximal functions. As one particular consequence of the estimates he obtains a new lower bound for the Hausdorff dimension \(\ell\) of the Kakeya set in \(\mathbb{R}^d\), namely \(\ell\geq (d+ 2)/2\).