Recall that if a lattice is described by lattice vectors \(\vec a_1, \vec a_2, \vec a_3\), the reciprocal lattice vectors are defined by the following 9 constraints: \begin{equation*} \vec a_i \cdot \vec b_j = 2\pi \delta_{ij} \end{equation*}
On this reciprocal lattice, the Brillouin zone is defined to be the set of all points closer to the origin that any other lattice points.