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Octahedral norms and convex combination of slices in Banach spaces. (English) Zbl 1297.46011

A number of papers over the last 15 years have shown that Banach spaces \(X\), where all relatively weakly open subsets of the unit ball \(B_X\) have diameter two, form a rich and interesting class of Banach spaces. We say that these spaces enjoy the diameter 2 property, D2P for short. Only recently, the present authors have shown that the D2P is strictly stronger than “every slice of \(B_X\) having diameter 2”. The paper under review concerns a proper subclass of D2P-spaces, namely those spaces for which every convex combination of slices (or, equivalently, relatively weakly open subsets) of \(B_X\) has diameter two. We say that such spaces have the strong D2P (SD2P).
The paper starts with a complete proof that \(X\) has an octahedral norm if and only if every convex combination of weak-star slices in \(B_{X^\ast}\) has diameter 2. This result is due to Deville and Godefroy in the late 1980s (see page 12 of [G. Godefroy, Stud. Math. 95, No. 1, 1–15 (1989; Zbl 0698.46011)]), but a complete proof has not been published till now. (This result is also proved in [R. Haller, J. Langemets and M. Põldvere, “On duality of diameter 2 properties”, to appear in J. Convex Anal. 22, No. 2 (2015), arxiv:1311.2177], together with corresponding results for D2P and its slice version.) Anyway, the point is that we now get that \(X\) has the SD2P if and only if the norm of \(X^\ast\) is octahedral.
An interesting question is “What are the Banach spaces such that both \(X\) and \(X^\ast\) have the SD2P?”. In [T. Abrahamsen et al., J. Convex Anal. 20, No. 2, 439–452 (2013; Zbl 1274.46027)] it was observed that spaces with the Daugavet property belong to this class. In the paper under review, this observation is considerably improved: Spaces with the almost Daugavet property are also included.
In [Godefroy, loc. cit.] it was asked if every \(X\) containing \(\ell_1\) can be equivalently renormed so that the bidual norm is octahedral. This is equivalent to the question whether every such \(X\) can be equivalently renormed so that every convex combination of slices in \(B_{X^\ast}\) has diameter 2. The authors prove, in Proposition 2.11, that one can renorm in the separable case to obtain at least \(2-\varepsilon\), for every \(\varepsilon>0\).

MSC:

46B04 Isometric theory of Banach spaces
46B03 Isomorphic theory (including renorming) of Banach spaces
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References:

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