Moduli of non-dentability and the radon-nikodým property in banach spaces |
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Authors: | W Schachermayer A Sersouri E Werner |
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Institution: | (1) Johannes Kepler University, Linz, Austria;(2) Equipe d’Analyse, Université Paris VI and University of Texas, Austin;(3) Equipe d’Analyse, Univesité Paris VI and Oklahoma State University, Stillwater |
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Abstract: | We show that for a separable Banach spaceX failing the Radon-Nikodym property (RNP), andε > 0, there is a symmetric closed convex subsetC of the unit ball ofX such that every extreme point of the weak-star closure ofC in the bidualX** has distance fromX bigger than 1 −ε. An example is given showing that the full strength of this theorem does not carry over to the non-separable case. However,
admitting a renorming, we get an analogous result for this theorem in the non-separable case too. We also show that in a Banach
space failing RNP there is, forε > 0, a convex setC of diameter equal to 1 such that each slice ofC has diameter bigger than 1 −ε. Some more related results about the geometry of Banach spaces failing RNP are given. |
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