Abstract :
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 cn(X) of a finite-dimensional Banach space X,
and in particular of a Hilbert space. Note that cn(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 cn(H) = nn/2. The same value of cn(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 dn/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 dimH = 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 cn(H) and c(H). In
particular, we answer a question raised by J.C. García-Vázquez and R. Villa [Mathematika 46 (1999)315–322]. Also, we prove the conjectured cn(H) = nn/2 for real Hilbert spaces of dimension n 5.
A few further estimates have been also derived.
2004 Elsevier Inc. All rights reserved.
