Trilinear Kakeya inequality in three dimensions
= Trilinear Kakeya inequality in three dimensions
For three transverse families of $R^{1/2}\times R^{1/2}\times R$ tubes in $\mathbb R^3$,
$$
\left\|\prod_{j=1}^3\left(\sum_{T\in\mathcal T_j}\chi_T\right)^{1/3}\right\|_{3/2}
\lesssim_\epsilon R^{1+\epsilon}\prod_{j=1}^3|\mathcal T_j|^{1/3}.
$$