On the moduli of convexity

Abstract

It is known that, given a Banach space (X,∥ · ∥), the modulus of convexity associated to this space δx is a non-negative function, nondecreasing, bounded above by the modulus of convexity of any Hilbert space and satisfies the equation δx(ε) /ε2 ≤ 4L δx(mu;)/μ2 for every 0 < ε ≤ μ ≤ 2, where L > 0 is a constant. We show that, given a function f satisfying these properties then, there exists a Banach space in such a way its modulus of convexity is equivalent to f, in Figiel's sense. Moreover this Banach space can be taken to be two-dimensional. © 2007 American Mathematical Society.