Journal of Convex Analysis 22 (2015), No. 2, 465--483
Copyright Heldermann Verlag 2015
On Duality of Diameter 2 Properties
Rainis Haller
Institute of Mathematics, University of Tartu, J. Liivi 2, 50409 Tartu, Estonia
rainis.haller@ut.ee
Johann Langemets
Institute of Mathematics, University of Tartu, J. Liivi 2, 50409 Tartu, Estonia
johann.langemets@ut.ee
Märt Põldvere
Institute of Mathematics, University of Tartu, J. Liivi 2, 50409 Tartu, Estonia
mart.poldvere@ut.ee
It is known that a Banach space has the strong diameter 2 property (i.e. every convex combination of slices of the unit ball has diameter 2) if and only if the norm on its dual space is octahedral (a notion introduced by Godefroy and Maurey). We consider two more versions of octahedrality, which are dual properties to the diameter 2 property and its local version (i.e., respectively, every relatively weakly open subset and every slice of the unit ball has diameter 2). We study stability properties of different types of octahedrality, which, by duality, provide easier proofs of many known results on diameter 2 properties.
Keywords: Diameter 2 property, slice, relatively weakly open set, octahedral norm.
MSC: 46B20, 46B22
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