Doctoral defence: Stefano Ciaci “Transfinite geometric properties of the unit ball in Banach spaces”

On 28 August at 14:00, Stefano Ciaci will defend his thesis “Transfinite geometric properties of the unit ball in Banach spaces“ for obtaining the degree of Doctor of Philosophy in Mathematics.

Supervisors:
Associate Professor Johann Langemets, University of Tartu
PhD Aleksei Lissitsin, University of Tartu

Opponent:
Professor Vladimir Kadets (Holon Institute of Technology, Israel)

Summary
The structure of the unit ball plays an important role in understanding the geometry of Banach spaces. At most, the maximal diameter of a subset of the unit ball in a Banach space can be two. Special interest has been turned to determining the diameter of slices, which are formed by intersecting of the unit ball with half-spaces. If every slice of the unit ball of a Banach space has diameter two, then the space is said to have the local diameter two property. This extreme geometric phenomenon can only happen in infinite-dimensional Banach spaces. A systematic treatment of diameter two properties and their relatives was started by T. A. Abrahamsen, V. Lima, and O. Nygaard in 2013. From there on, various strengthenings of diameter two properties and their relatives have emerged, e.g., almost square spaces and octahedral norms.

A common property for all of these notions above is that they are finitely defined; that is, the definitions require that for every finite number of elements in a Banach space or its dual, there is some special element in the space or the dual. Such geometric properties were formalized by J. D. Hardtke as test families in 2020. Surprisingly, many classical Banach spaces enjoy even a transfinite analogue of the diameter two properties or their relatives.

The main aim of the thesis is to systematically study transfinite extensions of the classical diameter two properties, almost squareness, and octahedral norms. The transfinite case generally exhibits significantly distinct behavior and is technically more complex. Hence, it gives new fruitful results and examples that substantially complement the existing theory of spaces with the diameter two properties and their relatives.