This paper has been motivated by previous work on estimating lower bounds for the norms of homogeneous polynomials which are products of linear forms. The purpose of this work is to investigate the so-called nth (linear) polarization constant c n (X) of a finite-dimensional Banach space X, and in particular of a Hilbert space. Note that c n (X) is an isometric invariant of the space. It has been proved by J. Arias-de-Reyna [Linear Algebra Appl. 285 (1998) 395-408] that if H is a complex Hilbert space of dimension at least n, then c n (H ) = n n/2 . The same value of c n (H ) for real Hilbert spaces is only conjectured, but estimates were obtained in many cases. In particular, it is known that the nth (linear) polarization constant of a d-dimensional real or complex Hilbert space H is of the approximate order d n/2 , for n large enough, and also an integral form of the asymptotic quantity c(H ), that is the (linear) polarization constant of the Hilbert space H , where dim H = d, was obtained together with an explicit form for real spaces. Here we present another proof, we find the explicit form even for complex spaces, and we elaborate further on the values of c n (H ) and c(H ). In particular, we answer a question raised by J.C. García-Vázquez and R. Villa [Mathematika 46 (1999) ✩

If L = 0 is a continuous symmetric n-linear form on a Banach space and L is the associated continuous n-homogeneous polynomial, the ratio L / L always lies between 1 and n n /n!. At one extreme, if L is defined on Hilbert space, then L / L = 1. If L attains norm on Hilbert space, then L also attains norm; in this case, we give an explicit construction to provide a unit vector x 0 with L = | L(x 0 )| = L . At the other extreme, if L / L = n n /n! and L attains norm, then L attains norm. We prove that in general the converse is not true.

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