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 2 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 .