On 28 August at 14:30 Rihhard Nadel will defend his doctoral thesis “Big slices of the unit ball in Banach spaces” for obtaining the degree of Doctor of Philosophy (Mathematics).
Senior Research Fellow Rainis Haller, PhD, University of Tartu
Research Fellow Johann Langemets, PhD, University of Tartu
Assoviate Professor Vegard Lima, Dr Scient., University of Agder (Norway)
Professor Vladimir Kadets, DSc, Kharkiv V. N. Karazin National University (Ukraine)
Assistant Professor Antonín Procházka, PhD, University of Franche-Comté (France)
Lately, in the field of the geometry of Banach spaces, a large amount of attention has been devoted to the study of diameter 2 properties which describe Banach spaces in which every slice of the unit ball has diameter 2. For example, the classical Banach spaces c0, ℓ∞, C[0,1], L1[0,1] and L∞[0,1] all have this property. On the other hand, in reflexive spaces, for example Hilbert spaces, or in separable dual spaces, for example ℓ1, there are slices with arbitrarily small diameter and thus they fail to have the diameter 2 properties. A systematic treatment of diameter 2 properties was started by T. A. Abrahamsen, V. Lima, and O. Nygaard in 2013. The aim of the thesis is to study strengthenings of the classical diameter-2 properties and related notions, such as roughness of the norm and the Daugavet indices of thickness. We give a complete description of how different versions of strong diameter 2 properties, rough norms and the Daugavet indices of thickness behave on absolute norms. The inheritance of these properties from a space to its subspaces, and vice versa, is also investigated. We study what criteria must a metric space meet in order for the space of Lipschitz functions on the metric space to have the w*-symmetric strong diameter 2 property. We quantitatively generalise known roughness results for spaces of bounded linear operators. As a result of our study of the Daugavet index of thickness, we give a negative answer to a question posed by Y. Ivakhno in 2006 about the relationship between the local diameter 2 property and the r-big slice property.