Kakeya Minkowski dimension conjecture (${\dim }_{\mathrm{M}}E=n$)

The dimension assertion asks that every bounded Kakeya set in ${\mathbb{R}}^n$ have full Minkowski dimension. The Kakeya maximal conjecture implies the stronger lower-dimension conclusion: its application to $E_{\delta }$ gives $|E_{\delta }|{\gtrsim }_{n,\epsilon }{\delta }^{n\epsilon }$, hence $N_{\delta }(E){\gtrsim }_{n,\epsilon }{\delta }^{-n+n\epsilon }$ for every $\epsilon >0$.