Another look at the continuity of moduli of noncompact convexity

TL;DR

This paper introduces a new modulus of noncompact convexity and proves its continuity for minimalizable measures of noncompactness on metric spaces, extending classical results.

Contribution

It defines a new modulus of noncompact convexity and establishes its continuity under broad conditions, advancing the understanding of convexity measures.

Findings

The new modulus differs from the classical one in general cases.

Continuity of the modulus is proven for minimalizable measures.

Results apply to arbitrary metric spaces.

Abstract

In this article we introduce the new modulus $\triangle'_{X,\phi}(\varepsilon)$, for which we prove that in the general case is different from the classical modulus of noncompact convexity. The main result of the paper is showing the continuity of the modulus of noncompact convexity for arbitrary minimalizable (strictly minimalizable) measure of noncompactness on arbitrary metric space.