Dimensionality reduction in extremal problems for moments of linear combinations of vectors with random coefficients

It is shown that sup{Ephi (\\Sigma(i=1)(n) xiiXi\\(2)) : x(i) epsilon H, Sigma(i=1)(n)\\x(i)\\(2) = B-2} does not depend on dim H greater than or equal to 1, where (H, \\(.)\\) is a Hilbert space, W is any convex function, and are any (real-valued) random variables. An immediate corollary is the following vector extension of the Whittle-Haagerup inequality: let xi(1),...,xi(n) be independent Rademacher random variables, and let x1,..., x, be vectors in H; then E \\Sigma(i=1)(n) epsilon(i)x(i)\\(p) less than or equal to E\v\(p) (Sigma(i=1)(n) \\x(i)\\(2))(p/2) For Allp greater than or equal to 2, where v similar to N(0, 1). Dimensionality reduction in the case when all the lengths \\x(i)\\ are fixed is also considered. Open problems are stated.