Granero, A. S.; Jiménez-Sevilla, M.; Moreno, J. P. Intersections of closed balls and geometry of Banach spaces. (English) Zbl 1069.46007 Extr. Math. 19, No. 1, 55-92 (2004). This well-written survey article on the Mazur intersection property (MIP) and its variants, by three well-known experts of this field, contains a very up-to-date and lucid account of some of the recent developments in this theory. We will be focussing only on results related to the MIP in this review. We recall that a Banach space has the MIP if every closed and bounded set can be represented as an intersection of closed balls. A recent key result due to D. Chen and B.-L. Lin [Bull. Pol. Acad. Sci., Math. 43, No. 3, 191–198 (1995; Zbl 0836.46009)] combined with earlier work of J. R. Giles, D. A. Gregory and B. Sims [Bull. Aust. Math. Soc. 18, 105–123 (1978; Zbl 0373.46028)] characterizes the MIP in terms of the geometry of the dual unit ball as “\(X\) has the MIP if and only if there is a dense set of semi-denting points or a dense set of weak\(^{\ast }\) denting points”.A wide class of Banach spaces admit equivalent norms with the MIP. These ideas are discussed in the second and third section of the paper. By a result of R. Deville, G. Godefroy and V. Zizler [Mathematik 40, No. 2, 305–321 (1993; Zbl 0792.46007)], any Banach space that has the Radon Nikodym property and a Fréchet differentiable bump function has such an equivalent norm. The space constructed by S. Shelah [Isr. J. Math. 51, 273–297 (1985; Zbl 0589.03012)] does not admit an equivalent norm with the MIP.Let \(M\) denote the collection of all intersections of balls. The porosity aspects of \(M\) w.r.t. the Hausdorff metric are discussed in section 4. A Banach space fails the MIP if and only if \(M\) is uniformly very porous. The stability aspects of \(M\) and the class of Mazur sets is the content of sections 5 and 6. For instance, \(M\) is stable under vector sums but not necessarily under intersections. Mazur spaces are those in which any intersection of balls is a Mazur set. For any set \(I\), \(c_{0}(I)\) is a Mazur space. A Banach space has dimension less than three if and only if it is a Mazur space w.r.t. every equivalent norm. Reviewer: T.S.S.R.K. Rao (Bangalore) Cited in 1 ReviewCited in 11 Documents MSC: 46B20 Geometry and structure of normed linear spaces 46-02 Research exposition (monographs, survey articles) pertaining to functional analysis Keywords:Mazur intersection property; Mazur space Citations:Zbl 0836.46009; Zbl 0373.46028; Zbl 0792.46007; Zbl 0589.03012 PDFBibTeX XMLCite \textit{A. S. Granero} et al., Extr. Math. 19, No. 1, 55--92 (2004; Zbl 1069.46007) Full Text: EuDML