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

We show that for a separable Banach space X failing the Radon-Nikodým property (RNP), and ε > 0, there is a symmetric closed convex subset C of the unit ball of X such that every extreme point of the weak-star closure of C in the bidual X** has distance from X 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, for ε > 0, a convex set C of diameter equal to 1 such that each slice of C has diameter bigger than 1 -ε. Some more related results about the geometry of Banach spaces failing RNP are given.