Continuity of modulus of noncompact convexity for minimalizable measures of noncompactness

Abstract

We consider the modulus of noncompact convexity ${\Delta }_{X,\varphi }(\epsilon )$ associated with the minimalizable measure of noncompactness $\varphi$. We present some properties of this modulus, while the main result of this paper is showing that ${\Delta }_{X,\varphi }(\epsilon )$ is a subhomogenous and continuous function on $[0,\varphi ({\bar{B}}_X))$ for an arbitrary minimalizable measure of compactness $\varphi$ in the case of a Banach space $X$ with the Radon-Nikodym property.