Renorming Banach spaces with the Mazur intersection property (Q676205)

A Banach space \((X,\| \;.\;\|)\) has the Mazur intersection property if every closed convex bounded subset of \(X\) is the intersection of all balls containing it; this can be understood as a smoothness property of the norm \(\| \;.\;\| \). It has been asked by \textit{J. R. Giles, D. A. Gregory} and \textit{B. Sims} [Bull. Aust. Math. Soc. 18, 105-123 (1978; Zbl 0373.46028)] whether a space with the Mazur intersection property is actually an Asplund space. Among the many interesting results of this paper is a negative answer to this question, since the authors show in Corollary 2.8 that every Banach space can be isometrically embedded into a space with the Mazur intersection property. In another direction it is shown that for each tree \(T\), the space \(C_{0}(T)\) has an equivalent norm with the Mazur intersection property, but a certain Banach space constructed by Kunen using the continuum hypothesis is an example of an Asplund space having no equivalent norm with the Mazur intersection property; in fact, it doesn't even have a nicely smooth norm. In addition, the three-space problem for the Mazur intersection property is studied.