MAZUR INTERSECTION PROPERTIES AND DIFFERENTIABILITY OF CONVEX FUNCTIONS IN BANACH SPACES P. G. GEORGIEV, A. S. GRANERO, M. JIME  NEZ SEVILLA  J. P. MORENO A It is proved that the dual of a Banach space with the Mazur intersection property is almost weak* Asplund. Analogously, the predual of a dual space with the weak* Mazur intersection property is almost Asplund. Through the use of these arguments, it is found that, in particular, almost all (in the Baire sense) equivalent norms on   (Γ) and   (Γ) are Fre  chet differentiable on a dense G δ subset. Necessary conditions for Mazur intersection properties in terms of convex sets satisfying a Krein–Milman type condition are also discussed. It is also shown that, if a Banach space has the Mazur intersection property, then every subspace of countable codimension can be equivalently renormed to satisfy this property. 1. Introduction Would you say that almost all (in the Baire sense) equivalent norms on   (Γ) and   (Γ) are Fre  chet differentiable on a dense G δ subset? The main goal of this paper is to show that this is the case, despite the (traditionally assumed) bad behaviour of these spaces regarding differentiability. These results are derived from a more general discussion of the geometric nature of Mazur’s intersection properties. It was Mazur [14] who began the study of determining those normed linear spaces which have what came to be known as the Mazur intersection property : every bounded closed convex set is an intersection of closed balls. Phelps [18] and Sullivan [23] pursued this study and, finally, Giles, Gregory and Sims found very useful characterizations [7]. They also considered, for a dual space, the weak* Mazur intersection property : every bounded weak* closed convex set is an intersection of closed dual balls. After these pioneering works, a good deal of attention has been paid to these (and other) ball separation properties (see [2, 3] and references therein). The connections between the Mazur intersection property and differentiability of convex functions were already noticed by Mazur and later investigated by Kenderov and Giles [12], Georgiev [5] and Moreno [16], among others. It has been recently proved, however, that there are no relationships between Asplund spaces and spaces with the Mazur intersection property [10]. As a matter of fact, the geometric intuition associated to Asplund spaces is of somewhat limited usefulness when dealing with Mazur and weak* Mazur intersection properties. For instance, it is worth emphasizing that a dual space possesses nice differentiability properties when the predual has the Mazur intersection property. Actually, this property implies a conexity condition for the dual unit ball and so we would expect to obtain differentiability in the predual. We have an analogous situation when the dual has the Received 23 July 1998 ; revised 14 December 1998. 2000 Mathematics Subject Classification 46B20. Supported in part by Bulgarian National Foundation for Scientific Investigations grant MM 70395 and by DGICYT grant PB 94-0243 and PB 96-0607. J. London Math. Soc. (2) 61 (2000) 531–542