Linear polarization constant of $\R^n$

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.