Mazur’s Intersection Property and its Variants

Abstract

In this thesis, we discuss various differentiability notions in connection with ball separation prop- erties. We characterise the uniform Mazur’s intersection property (UMIP) in terms of w*-semidenting points in attempt to resolve a long standing open question: “Does UMIP imply uniformly smooth renorming?” Further, we discuss a stronger version of UMIP called the hyperplane uniform Mazur intersection property (HUMIP) which is shown to characterise uniform smoothness. Similar ball separation char- acterisations are obtained for Fr´echet smoothness and asymptotic uniform smoothness (AUS). These ball separation properties are then shown to be residual properties. Thus, we obtain that norms which have UMIP or norms which are (asymototically) uniformly smooth are residual in the set of all equivalent norms. This also helps taking the open question forward which asks for residuality of Fr´echet smooth norms. Also, in attempt to understand UMIP better, we discuss UMIP from some quantitative aspects. We obtain conditions for the stability of UMIP under ℓp-sum and use an example by Borwein and Fabian to answer the following question in negative: “Does hereditary MIP imply Fr´echet smooth- ness?” Some interesting problems and possible approaches are discussed at the end.