An extension of a Bourgain--Lindenstrauss--Milman inequality

Let || . || be a norm on R^n. Averaging || (\eps_1 x_1, …, \eps_n x_n) || over all the 2^n choices of \eps = (\eps_1, …, \eps_n) in {-1, +1}^n, we obtain an expression ||| . ||| which is an unconditional norm on R^n. Bourgain, Lindenstrauss and Milman showed that, for a certain (large) constant \eta > 1, one may average over (\eta n) (random) choices of \eps and obtain a norm that is isomorphic to ||| . |||. We show that this is the case for any \eta > 1.