An optimal plank theorem
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- by Oscar Ortega-Moreno PDF
- Proc. Amer. Math. Soc. 149 (2021), 1225-1237 Request permission
Abstract:
We give a new proof of Fejes Tóth’s zone conjecture: for any sequence $v_1,v_2,\dots ,v_n$ 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 a 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.References
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Additional Information
- Oscar Ortega-Moreno
- Affiliation: Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien Wiedner Hauptstraße 8–10/104, 1040 Wien, Austria
- Email: oscarortem@gmail.com
- Received by editor(s): October 25, 2019
- Received by editor(s) in revised form: March 9, 2020, and April 16, 2020
- Published electronically: January 13, 2021
- Additional Notes: The author was supported by the Austrian Science Fund (FWF), Project number: P31448-N35, and the Mexican National Council of Science and Technology (CONACYT), grant number: CVU579817
- Communicated by: Deane Yang
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 149 (2021), 1225-1237
- MSC (2020): Primary 52C99, 46C05
- DOI: https://doi.org/10.1090/proc/15228
- MathSciNet review: 4211876