The complex plank problem, revisited

Abstract: Ball's complex plank theorem states that if $v_1,\dots,v_n$ are unit vectors in $\mathbb{C}<sup>d$, and $t_1,\dots,t_n$, non-negative numbers satisfying $\sum_{k=1}^nt_k² = 1,$ then there exists a unit vector $v$ in $\mathbb{C}<sup>d$ for which $|\langle v_k,v \rangle | \geq t_k$ for every $k$. Here we present a streamlined version of Ball's original proof.