A Continuity Property Related to Kuratowski′s Index of Non-compactness, Its Relevance to the Drop Property, and Its Implications for Differentiability Theory

We define and study the properties of α upper semi-continuity, a new continuity property for set-valued mappings from a topological space into subsets of a metric space, expressed in terms of Kuratowski′s index of non-compactness. This α upper semi-continuity is related closely to the usual upper semi-continuity but significantly α upper semi-continuous minimal weak* cuscos from a Baire space into subsets of the dual of any Banach space are generically single-valued. Kuratowski′s index of non-compactness has been used to study the drop property and α upper semi-continuity is dual to property α studied there. Uniform α upper semi-continuity of the duality mapping is dual to nearly uniform rotundity properties. Importantly, α upper semi-continuity has application in differentiability theory, providing another characterisation for Asplund spaces. An examination of an associated concept, α-denting point for sets, yields still further advances concerning the differentiability of convex functions on a large class of Banach spaces