  The present work contributes to the determination of the $n$-th linear
polarization constant $c_n(H)$ of an $n$-dimensional real Hilbert space $H$. We
provide some new lower bounds on the value of $\sup_{\|y\|=1}| x_1,y >... x_n,y
|$, where $x_1, ..., x_n$ are unit vectors in $H$. In particular, the results
improve an earlier estimate of Marcus. However, the intriguing conjecture
$c_n(H)=n^{n/2}$ remains open.
