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.
Categories: math.CA