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 cn of Rn 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∥y∥=1∣〈x1, y〉 ⋯ 〈xn, y〉∣, where x1, … , xn are unit vectors in Rn. The new estimate is given in terms of the eigenvalues of the Gram matrix [〈xi, xj〉] and improves upon earlier estimates of this kind. However, the intriguing conjecture cn = nn/2 remains open.