Some properties of Gurarii's modulus of convexity

Abstract.

In 1967 Gurarii introduces a modulus of convexity, $\beta _{E}(\varepsilon)$. In this paper some properties of this modulus are analyzed. They have their analogous on the well known- Clarkson's modulus of convexity, $\delta _{E}(\varepsilon)$. In particular we show that $\beta _{E}(\varepsilon)$ can be defined in several equivalent forms. We also prove that in l_p spaces with 2 < p the Gurarii's modulus satifies the equality $\beta _{l_p}(\varepsilon)=1-\root p \of{1-\left({\varepsilon} \over {2} \right)^p}$ for $0\le \varepsilon \le 2$.