The complex plank problem, revisited

TL;DR

This paper revisits Ball's complex plank theorem, providing a streamlined proof and exploring the conditions under which a unit vector in complex space satisfies certain inner product bounds with given vectors.

Contribution

The paper offers a simplified proof of Ball's complex plank theorem, enhancing understanding and accessibility of this important result in complex vector spaces.

Findings

Streamlined proof of Ball's complex plank theorem

Clarification of conditions for existence of a vector with prescribed inner product bounds

Potential implications for complex vector space geometry

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 $\sum_{k=1}^nt_k^2 = 1,$ then there exists a unit vector $v$ in $\mathbb{C}^d$ for which $|\langle v_k,v \rangle | \geq t_k$ for every $k$. Here we present a streamlined version of Ball's original proof.