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First Borel class sets in Banach spaces and the asymptotic-norming property. (English) Zbl 1065.46010
This article deals with Banach spaces which are Borel sets of the first class in their biduals equipped with the weak-star topology, and contains in particular extensions of results due to G. A. Edgar and R. F. Wheeler [Pac. J. Math. 115, 317–350 (1984; Zbl 0506.46007))] and N. A. Ghoussoub and B. Maurey [J. Funct. Anal. 61, No. 1, 72–97 (1985; Zbl 0565.46011)); Proc. Am. Math. Soc. 94, No. 4, 665–671 (1985; Zbl 0587.46015); Mem. Am. Math. Soc. 62, No. 349 (1986; Zbl 0606.46005)]. Edgar and Wheeler showed that when the complement of a Banach space \(X\) in its bidual is a countable union of weak-star compact convex sets, then \(X\) has the Radon–Nikodym property. The converse, however, fails. It is shown in the article under review that the equivalence holds true for separable spaces \(X\) if one replaces “weak-star compact convex sets” by “differences of two weak-star compact convex sets”.
Moreover, this representation can be obtained with differences which are at positive distance (in norm) from \(X\) if and only if there is an equivalent norm on \(X\) with the asymptotically norming property, without assuming separability of \(X\). Other interesting renorming results are proved: for instance, an arbitrary Banach space has an equivalent locally uniformly rotund norm if and only if every norm open subset is a countable union of differences of closed convex sets. Let us mention that it is apparently still unknown whether every Banach space with the Radon–Nikodym property has an equivalent locally uniformly rotund norm.

MSC:
46B03 Isomorphic theory (including renorming) of Banach spaces
46B20 Geometry and structure of normed linear spaces
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
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