0n June 5th at 16.15 Jaan Kristjan Kaasik will defend his thesis „Diameter two properties and almost square properties in Lipschitz-free spaces and spaces of Lipschitz functions“. The defense will take place in the Delta study building, Narva mnt 18-1020.
Supervisors:
Professor of Mathematical Analysis, Academician Rainis Haller, University of Tartu
Research Fellow in Mathematical Analysis Andre Ostrak, University of Tartu
Opponent:
Associate Professor Marek Cuth, Charles University in Prague (Czech Republic)
Summary
Diameter two properties and almost square properties in Lipschitz-free spaces and spaces of Lipschitz functions Diameter two properties and almost square properties describe the geometric shape of the unit ball of a Banach space. In particular, they express whether certain subsets of the unit ball necessarily have the maximal possible diameter. These properties constitute an important direction in the study of the geometry of Banach spaces.
This thesis investigates diameter two properties and almost square properties in spaces of Lipschitz functions and in their canonical preduals, the Lipschitzfree spaces. It is shown that a Lipschitz-free space over a length metric space is locally almost square. On the other hand, it is proven that no Lipschitz-free space is almost square. Moreover, an example of a Lipschitz-free space that is locally almost square but not weakly almost square is constructed. The thesis also studies diameter two properties in spaces of Lipschitz functions and provides new metric characterisations of these properties. Using new examples, all diameter two properties are shown to differ in these spaces.
