Open Collections

An optimal plank theorem

Banff International Research Station for Mathematical Innovation and Discovery

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.

Mathematics; Convex and discrete geometry; Functional analysis

video/mp4

Author affiliation: University of Warwick

2020-08-12

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/75429

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/