Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The real plank problem and some applications


Authors: G. A. Muñoz-Fernández, Y. Sarantopoulos and J. B. Seoane-Sepúlveda
Journal: Proc. Amer. Math. Soc. 138 (2010), 2521-2535
MSC (2000): Primary 46G25; Secondary 51M16, 47H60
Published electronically: February 23, 2010
MathSciNet review: 2607882
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

$\displaystyle \left\vert\langle{x}, x_{k}\rangle\right\vert\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

$\displaystyle \left\vert\langle{x}, x_{k}\rangle\right\vert \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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46G25, 51M16, 47H60

Retrieve articles in all journals with MSC (2000): 46G25, 51M16, 47H60


Additional Information

G. A. Muñoz-Fernández
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza de las Ciencias 3, 28040, Madrid, Spain
Email: gustavo_fernandez@mat.ucm.es

Y. Sarantopoulos
Affiliation: Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University, Zografou Campus 157 80, Athens, Greece
Email: ysarant@math.ntua.gr

J. B. Seoane-Sepúlveda
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza de las Ciencias 3, 28040, Madrid, Spain
Email: jseoane@mat.ucm.es

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10295-0
PII: S 0002-9939(10)10295-0
Keywords: Plank problems, polarization constants, product of linear functionals.
Received by editor(s): November 6, 2009
Published electronically: February 23, 2010
Additional Notes: The first author was supported by MTM2006-03531.
The second author was partly supported by the National Technical University: 2007 basic research program ‘C. Carathéodory’, No. 65/1602
The third author was supported by MTM2006-03531.
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2010 American Mathematical Society