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Description

We give a new proof of Fejes Tóthâ s zone conjecture: for any sequence $v_1,v_2,...,v_n$ of unit vectors in a real Hilbert space $H$, there exists a unit vector $v$ in $H$ 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. As a result, we obtain a unified approach to some of the most important plank problems on the real and complex setting: the classic plank problem, the complex plank problem, Fejes Tóthâ s zone conjecture and the strong polarization problem.