Convexity, Optimization and Geometry of the Ball in Banach Spaces
The research done during my PhD involves several topics in the framework of the geometry of Banach spaces. The guiding theme is the structure of convex sets. Several properties based on the notions of extreme and exposed points are analyzed. This includes classical properties as the Radon-Nikodym property, which is studied from a non-linear point of view, as well as the study of the geometry of the norm in tensor products. Moreover, some isometric properties (in particular, the Daugavet property and the extremal structure) are studied in spaces of Lipschitz functions and their preduals. These last spaces, known as Lipschitz-free spaces, are a very active research topic by their applications in non-linear analysis. The research has been done in collaboration with the reasearch groups of Murcia and Granada (Spain) and Besançon (France). To this end, a three months research stay was done in Besançon, and a two-weeks research stay was done in Granada.
My thesis was supervised by Bernardo Cascales and Matías Raja.
(comics from xkcd and Spiked Math)