ON SOME OPEN PROBLEMS IN BANACH SPACE THEORY

The main line of investigation of the present work is the study of some aspects in the analysis of the structure of the unit ball of (infinite-dimensional) Banach spaces. In particular, we analyse some questions concerning the existence of suitable renormings that allow the new unit ball to possess a specific geometric property. The main part of the thesis is, however, dedicated to results of isometric nature, in which the original norm is the one under consideration. One of the main sources for the selection of the topics of investigation has been the recent monograph [GMZ16], entirely dedicated to collecting several open problems in Banach space theory and formulating new lines of investigation. We take this opportunity to acknowledge the authors for their effort, that offered such useful text to the mathematical community. The results to be discussed in our work actually succeed in solving a few of the problems presented in the monograph and are based on the papers [HáRu17, HKR18, HáRu19, HKR••]. Let us say now a few words on how the material is organised. The thesis is divided in four chapters (some whose contents are outlined below) which are essentially independent and can be read in whatsoever order. The unique chapter which is not completely independent from the others is Chapter 4, where we use some results from Chapter 2 and which is, in a sense, the non-separable prosecution of Chapter 3. However, cross-references are few (never implicit) and usually restricted to quoting some result; it should therefore be no problem to start reading from Chapter 4. The single chapters all share the same arrangement. A first section is dedicated to an introduction to the subject of the chapter; occasionally, we also present the proof of known results, in most cases as an illustration of an important technique in the area. In these introductions we strove to be as self-contained as possible in order to help the novel reader to enter the field; consequently, experts in the area may find them somewhat redundant and prefer to skip most parts of them. The first section of each chapter concludes with the statement of our most significant results and a comparison with the literature. The proofs of these results, together with additional results or generalisations, are presented in the remaining sections of the chapter. These sections usually follow closely the corresponding articles (carefully referenced) where the results were presented.