On the modulus of $U$-convexity

Abstract

We prove that the moduli of $U$-convexity, introduced by Gao (1995), of the ultrapower $\tilde{X}$ of a Banach space $X$ and of $X$ itself coincide whenever $X$ is super-reflexive. As a consequence, some known results have been proved and improved. More precisely, we prove that $u_X\left(1\right)>0$ implies that both $X$ and the dual space $X^{\ast }$ of $X$ have uniform normal structure and hence the “worth” property in Corollary 7 of Mazcuñán-Navarro (2003) can be discarded.