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 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‖y‖=1 | 〈x1,y〉 · · · 〈xn,y〉 |, where x1, . . . ,xn are unit vectors in R. 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 = n n/2 remains open. 2000 Mathematics Subject Classification. Primary 46G25; Secondary 52A40, 46B07.