Three characterizations of polyhedral Banach spaces
An infinite-dimensional Banach space E is called polyhedral, if intersections of all finite dimensional spaces with the unit ball are polyhedra. Three characterizations, up to isomorphism, of such spaces are given. A ``local'' characterization uses a normed set in the unit dual sphere. A ``global'' characterization uses special coverings of the underlying Banach space [cf., \textit{V. Klee}, Stud. Sci. Math. Hung., 21, 415-427 (1986; Zbl 0577.52007)]. An ``internal'' characterization uses covering of the unit ball U(E) by a sequence of vanishing slices \(\{x\in U(E):f_ n(x)\geq 1-\alpha_ n\}\), \(\alpha_ n\to 0\), where \(f_ n\) are norm-1 functionals.
scientific article; zbMATH DE number 4203056