The modulus of nondensifiable convexity and its applications

Abstract

Based in the so called degree of nondensifiability, denoted by $\phi$, we introduce and analyze the concepts of modulus of nondensifiable convexity and the nearly uniform convexity characteristic of a given Banach space $X$ associated to $\phi$, denoted by $\Delta_{X}^{\phi}$ and $\varepsilon^{\phi}(X)$, respectively. Although $\phi$ is not a measure of noncompactness (MNC), we prove that $\Delta_{X}^{\phi}$ and $\varepsilon^{\phi}(X)$ are, respectively, a lower and upper bound for the modulus of noncompact convexity and the nearly uniform convexity characteristic of $X$ associated to an arbitrary MNC. Also, we characterize the normal structure of $X$ in terms of $\phi$ and, by using $\varepsilon^{\phi}(X)$, we give a sufficient condition for $X$ has the weak fixed point property.