On the moduly of convexity

Abstract

[EN] It is known that, given a Banach space (X, parallel to center dot parallel to), the modulus of convexity associated to this space delta X is a non-negative function, nondecreasing, bounded above by the modulus of convexity of any Hilbert space and satisfies the equation delta x(epsilon)/epsilon(2) <= 4L delta x(mu)/mu(2) for every 0 < epsilon <= mu <= 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.