The real plank problem and some applications

Muñoz-Fernández, G.; Sarantopoulos, Y.; Seoane-Sepúlveda, J.

doi:10.1090/S0002-9939-10-10295-0

The real plank problem and some applications
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by G. A. Muñoz-Fernández, Y. Sarantopoulos and J. B. Seoane-Sepúlveda PDF

Proc. Amer. Math. Soc. 138 (2010), 2521-2535 Request permission

Abstract:

K. Ball has proved the “complex plank problem”: if $\left (x_{k}\right )_{k=1}^{n}$ is a sequence of norm $1$ vectors in a complex Hilbert space $\left (H, \langle \cdot ,\cdot \rangle \right )$, then there exists a unit vector $x$ for which \[ \left |\langle {x}, x_{k}\rangle \right |\geq 1/\sqrt {n} ,\quad k=1, \ldots , n . \] In general, this result is not true on real Hilbert spaces. However, in special cases we prove that the same result holds true. In general, for some unit vector $x$ we have derived the estimate \[ \left |\langle {x}, x_{k}\rangle \right | \geq \max \left \{\sqrt {\lambda _{1}/n}, 1/\sqrt {\lambda _{n}n}\right \} , \] where $\lambda _{1}$ is the smallest and $\lambda _{n}$ is the largest eigenvalue of the Hermitian matrix $A=\left [\langle {x_{j}}, x_{k}\rangle \right ]$, $j, k=1, \ldots , n$. We have also improved known estimates for the norms of homogeneous polynomials which are products of linear forms on real Hilbert spaces.

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