On the moduli of convexity

Abstract:

It is known that, given a Banach space $(X,\|\cdot \|)$, the modulus of convexity associated to this space $\delta _X$ is a non-negative function, non-decreasing, bounded above by the modulus of convexity of any Hilbert space and satisfies the equation $\frac {\delta _X(\varepsilon )}{\varepsilon ^2}\leq 4L\frac {\delta _X(\mu )}{\mu ^2}$ for every $0<\varepsilon \leq \mu \leq 2$, 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.