Geometric properties in Orlicz spaces, direct sums and Banach spaces of vector-valued functions

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Since James Clarkson introduced the notion of uniform convexity in 1936, the study of convexity and smoothness properties of Banach spaces forms an important subfield of functional analysis. Such properties are interesting both from a theoretical and a more applied point of view.A principal question of Banach space theory concerns the stability of these geometric properties with respect to typical constructions of functional analysis, which include in particular direct sums and spaces of vector-valued functions such as Orlicz-Bochner spaces or the more general Köthe-Bochner spaces.There exists already a wide literature on classical convexity and smoothness properties of such spaces. In my dissertation, I have conducted extensive studies on some generalised notions of convexity and smoothness (the so called acs properties) in direct sums and Köthe-Bochner spaces.A far more wide-reaching generalisation of both the concepts of direct sums and Köthe-Bochner spaces are the so called direct integrals of Banach spaces, which were introduced in 1991 by Haydon, Levy and Raynaud for the study of representations of so called random Banach spaces. So far, the geometry of these direct integrals has not been studied systematically.It is therefore the main aim of this project to conduct comprehensive research on convexity and smoothness properties of direct integrals. This requires also an explicit description of the dual space of a direct integral, which is of independent interest. Such a description is not known until now and thus it has to be developed within this project, building on the known duality theory for Köthe-Bochner spaces.It is planned to study classical properties (e. g. strict or uniform convexity, Gateaux- or Frechet-differentiability of the norm) as well as the above-mentioned acs properties and several other generalised notions of convexity in direct integrals. New results especially on the geometry of infinte direct sums and Orlicz-/Köthe-Bochner spaces are also expected.