A geometric estimate on the norm of product of functionals - Mate Matolcsi (arXiv.org, 2006)
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.