Recall that a Banach space X has the Radon–Nikodým (RN) property if and only if the Radon–Nikodým theorem holds for vector measures with values in X (see [12]). This property plays a central role in the theory of vector measures. It has been clear for a long time that dual Banach spaces with the Radon–Nikodým property and their weak compact subsets play special role in this theory (see [33],[37]). Namioka [26] defined a compact space to be Radon–Nikodým (RN) compact if and only if it is homeomorphic to a weak compact subset of a dual Banach space with the Radon–Nikodým property. For example, as reflexive Banach spaces are dual spaces with the Radon–Nikodým property, the results of [11] imply that Eberlein compact spaces are RN compact. In [26] Namioka proved a number of interesting properties of RN compacta while also giving them an elegant internal characterization. The investigation of this class of compact spaces continued later, with some remarkable results like the relation with Corson and Eberlein compact spaces (see [30],[38]). But the question which has attracted more attention and produced a larger literature on RN compacta is the following very basic problem, already posed in [26] and traced in [15]–[18], which has remained open up to this date.