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An optimal plank theorem
Proceedings of the American Mathematical Society ( IF 0.8 ) Pub Date : 2021-01-13 , DOI: 10.1090/proc/15228
Oscar Ortega-Moreno

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
$\displaystyle \vert\langle v_k,v \rangle \vert \geq \sin (\pi /2n)$

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.


中文翻译:

最优木板定理

摘要:我们给出了FejesTóth区域猜想的新证明:对于真实希尔伯特空间中任何单位矢量序列,都存在一个单位矢量,使得 $ v_1,v_2,\点,v_n $ $ \ mathcal {H} $$ v $ $ \ mathcal {H} $
$ \ displaystyle \ vert \ langle v_k,v \ rangle \ vert \ geq \ sin(\ pi / 2n)$

为了所有$ k $。这可以看作是对实际希尔伯特空间的板条定理的一个尖锐版本。我们的方法受到Ball解决复杂木板问题的启发,因此在同一方法下将复杂解决方案与实际解决方案统一起来。
更新日期:2021-02-08
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