# An Optimal Plank Theorem

## TL;DR

This paper provides a new, unified proof of the zone conjecture in real Hilbert spaces, establishing a sharp version of the plank theorem by leveraging techniques inspired by Ball's complex plank problem solution.

## Contribution

It introduces a novel proof method that unifies real and complex cases of the plank theorem using insights from Ball's approach.

## Key findings

- Proves the zone conjecture for real Hilbert spaces.
- Establishes a sharp bound for the existence of a unit vector with specified inner product properties.
- Unifies real and complex plank problem solutions under a common framework.

## Abstract

We give a new proof of Fejes T\'oth's zone conjecture: for any sequence $v_1,v_2,...,v_n$ of unit vectors in a real Hilbert space $\mathcal{H}$, there exists a unit vector $v$ in $\mathcal{H}$ such that \begin{equation*} |\langle v_k,v \rangle| \geq \sin(\pi/2n) \end{equation*} for all $k$. This can be seen as sharp version of the plank theorem for real Hilbert spaces. Our approach is inspired by Ball's solution to the complex plank problem and thus unifies both the complex and the real solution under the same method.