A geometric estimate on the norm of product of functionals

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.