  The open problem of determining the exact value of the $n$-th linear
polarization constant $c_n$ of $\R^n$ has received considerable attention over
the past few years. This paper makes a contribution to the subject by providing
a new lower bound on the value of
  $\sup_{\|{\bf{y}}\|=1}| {\bf{x}}_1,{\bf{y}} ... {\bf{x}}_n,{\bf{y}}
  |$, where ${\bf{x}}_1, ... ,{\bf{x}}_n$ are unit vectors in $\R^n$. The new
estimate is given in terms of the eigenvalues of the Gram matrix $[
{\bf{x}}_i,{\bf{x}}_j ]$ and improves upon earlier estimates of this kind.
However, the intriguing conjecture $c_n=n^{n/2}$ remains open.
