Modulus of Convexity, the Coeffcient R ( 1 , X ) , and Normal Structure in Banach Spaces

Abstract

Let ${\delta }_{\text{X}}\left(ϵ\right)$ and $R\left(1,X\right)$ be the modulus of convexity and the Domínguez-Benavides coefficient, respectively. According to these two geometric parameters, we obtain a sufficient condition for normal structure, that is, a Banach space $X$ has normal structure if $2{\delta }_{\mathrm{X}}(1+ϵ)>\text{max}\left\{\left(\mathrm{R}\left(1,\mathrm{x}\right)-1\right)ϵ,1-\left(1-ϵ/R\left(1,X\right)-1\right)\right\}$ for some $ϵ\in \left[0,1\right]$ which generalizes the known result by Gao and Prus.