DAN

Abstract

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 the given vectors in the sense that $\prod |(u_i, v)| \geq n^{-n/2}$. The strong polarisation problem asserts that there is choice of $v$ for which $\sum 1/ (u_i, v)^2 \leq n^2$ holds. These follow from the complex plank problem if $H$ is a complex Hilbert space, but for real Hilbert spaces the general conjectures are still open. We prove special cases by transforming the statements to geometric forms and introducing inverse eigenvectors of positive semi-definite matrices.