The dth linear polarization constant of Rd

In [C. Benítez, Y. Sarantopoulos, A. Tonge, Lower bounds for norms of products of polynomials, Math. Proc. Cambridge Philos. Soc. 124 (3) (1998) 395–408] it was conjectured that for all unit vectors u1, . . . , ud in Rd , X (u1, . . . , ud ) := sup x∈Rd , |x|2=d d ∏ i=1 〈x,ui〉 1 with equality occurring iff u1, . . . , ud are orthonormal. We relate this to a conjecture about solutions of Ay = y−1, where A= (〈ui, uj 〉), and show that if the conjecture fails then the u1, . . . , ud minimizing X must be linearly dependent. We also show X (u1, . . . , ud) 1 for certain families of u1, . . . , ud . © 2008 Elsevier Inc. All rights reserved.