Octahedral norms and convex combination of slices in Banach spaces |
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Authors: | Julio Becerra Guerrero Ginés López-Pérez Abraham Rueda Zoca |
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Institution: | Universidad de Granada, Facultad de Ciencias, Departamento de Análisis Matemático, 18071-Granada, Spain |
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Abstract: | We study the relation between octahedral norms, Daugavet property and the size of convex combinations of slices in Banach spaces. We prove that the norm of an arbitrary Banach space is octahedral if, and only if, every convex combination of w?-slices in the dual unit ball has diameter 2, which answers an open question. As a consequence we get that the Banach spaces with the Daugavet property and its dual spaces have octahedral norms. Also, we show that for every separable Banach space containing ?1 and for every ε>0 there is an equivalent norm so that every convex combination of w?-slices in the dual unit ball has diameter at least 2−ε. |
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Keywords: | Slices Relatively weakly open sets Radon&ndash Nikodym property Renorming Octahedral norms |
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