An extension of a Bourgain–Lindenstrauss–Milman inequality

Let · be a norm on Rn. Averaging (ε1x1, . . . , εnxn) over all the 2n choices of −→ε = (ε1, . . . , εn) ∈ {−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 −→ε and obtain a norm that is isomorphic to ||| · |||. We show that this is the case for any η >1. © 2007 Elsevier Inc. All rights reserved