An extension of a Bourgain–Lindenstrauss–Milman inequality

AbstractLet ‖⋅‖ 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