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SUMMARY:Polarisation problems - Ambrus\, G (Renyi Institute)
DTSTART:20110302T151500Z
DTEND:20110302T161500Z
UID:TALK30085@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:Let $u_1\, u_2\, ...\, u_n$ be unit vectors in a Hilbert space
  $H$. The polarisation problem states that there is another unit vector $v
 $ in $H$\, which is sufficiently far from the orthogonal complements of th
 e given vectors in the sense that $prod |(u_i\, v)| geq n^{-n/2}$. The str
 ong polarisation problem asserts that there is choice of $v$ for which  $ 
 um 1/ (u_i\, v)^2 leq n^2$ holds. These follow from the complex plank prob
 lem if $H$ is a complex Hilbert space\, but for real Hilbert spaces the ge
 neral conjectures are still open. We prove special cases by transforming t
 he statements to geometric forms and introducing inverse eigenvectors of p
 ositive semi-definite matrices.\n\n\n
LOCATION:Seminar Room 1\, Newton Institute
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