The complex plank problem, revisited

Abstract

Ball's complex plank theorem states that if $v_1,\dots ,v_n$ are unit vectors in ${\mathbb{C}}^d$, and $t_1,\dots ,t_n$, non-negative numbers satisfying ${\sum }_{k=1}^n{t}_k^2=1,$ then there exists a unit vector $v$ in ${\mathbb{C}}^d$ for which $|⟨v_k,v⟩|\ge t_k$ for every $k$. Here we present a streamlined version of Ball's original proof.