An extension of a Bourgain-Lindenstrauss-Milman inequality

Let {norm of matrix} ṡ {norm of matrix} be a norm on Rⁿ. Averaging {norm of matrix} (ε₁ x₁, ..., ε_n x_n) {norm of matrix} over all the 2ⁿ choices of over(ε, →) = (ε₁, ..., ε_n) ∈ {- 1, + 1}ⁿ, we obtain an expression | | | x | | | which is an unconditional norm on Rⁿ. Bourgain, Lindenstrauss and Milman [J. Bourgain, J. Lindenstrauss, V.D. Milman, Minkowski sums and symmetrizations, in: Geometric Aspects of Functional Analysis (1986/1987), Lecture Notes in Math., vol. 1317, Springer, Berlin, 1988, pp. 44-66] showed that, for a certain (large) constant η > 1, one may average over ηn (random) choices of over(ε, →) and obtain a norm that is isomorphic to | | | ṡ | | |. We show that this is the case for any η > 1.