Moduli of non-dentability and the radon-nikodým property in banach spaces

Abstract

We show that for a separable Banach spaceX failing the Radon-Nikodým property (RNP), ande > 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 −e. 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, fore > 0, a convex setC of diameter equal to 1 such that each slice ofC has diameter bigger than 1 −e. Some more related results about the geometry of Banach spaces failing RNP are given.