Convexity, optimization and geometry of the ball in Banach spaces

Abstract

The aim of this dissertation is the study of several topics in the framework of the geometry of Banach spaces, with a focus on the structure of convex sets and its application in spaces of Lipschitz functions and their predual spaces. The methodology consisted on the study of the proposed topics with the supervisors and some specialists from other research groups. The student did a three-month stay in the Laboratoire de Mathématiques in Besançon under the supervision of Gilles Lancien. Moreover, he did two research visits, the first one in Granada supervised by Ginés López and the second one in Besançon supervised by Antonin Procházka. The student attended several conferences and schools in order to contact with international experts, be aware of recent research topics and present his progress. Now we summarise the content of the memory and the main results. Chapter 0 has an introductory character. It includes known results on the extremal structure, a general framework for set derivations and some results on tensor products. Chapter 1 deals with the class of compact convex sets which admit a strictly convex lower semicontinuous function. It is showed that a compact convex set belongs to that class if and only if it linearly embeds into a strictly convex dual Banach space endowed with the weak* topology. Moreover the existence of faces and exposed points of continuity for a strictly convex function is analysed. This chapter is based on a joint work with J. Orihuela and M. Raja. In Chapter 2, a version of the Radon-Nikodym property for maps is introduced. It is showed that the space of maps with such property inherits topological and geometrical properties from the target space. Moreover, it is studied the relation with the approximation by delta-convex maps and it is proved a version of Stegall's variational principle in this context. These results are part of a joint work with M. Raja. Chapter 3 deals with strong asymptotic uniform smoothness and convexity. These properties are used for providing a partial answer to the problem of whether the injective tensor product of AUS spaces is AUS. The results are applied in Orlicz spaces. Finally, it is showed that non-trivial injective tensor products are not strictly convex. These results were obtained in collaboration with M. Raja. Chapter 4 is devoted to the study of duality in spaces of vector-valued Lipschitz functions. In particular, the results of Weaver and Dalet have been generalised. Moreover, the notion of unconditional almost squareness is introduced. This notion is used for providing a partial answer to the question of the existence of dual ASQ spaces and for showing that some spaces of Lipschitz functions are not dual ones. This chapter is based on joint works with C. Petitjean and A. Rueda Zoca. Chapter 5 focuses on geometrical properties of Lipschitz free spaces and spaces of Lipschitz functions. It is showed that theses spaces have the Daugavet property if and only if the underlying metric space is a length space, which is a generalisation of a result by Ivakhno, Kadets and Werner. Moreover, it is studied the extremal structure of the ball of a free space. In particular, it is provided a characterisation of strongly exposed points. Finally, it is showed that Pelczynski's space is isomorphic to the free space on a compact convex set, and that it is not isomorphic to the free space over c0. This solves a problem posed by Cuth, Doucha and Wojtaszczyk. The results that appear in this chapter come from several papers with C. Petitjean, A. Procházka and A. Rueda Zoca.