Isomorphically polyhedral Banach spaces (Q2848856)

A real Banach space is polyhedral if the unit ball of each of its finite-dimensional subspaces is a polytope (see [\textit{V. P. Fonf, J. Lindenstrauss} and \textit{R. R. Phelps}, Infinite dimensional convexity. Handbook of the geometry of Banach spaces. Volume 1. Amsterdam: Elsevier, 599--670 (2001; Zbl 1086.46004), Section 6)]); it is isomorphically polyhedral if it is isomorphic to a polyhedral space. The authors prove two theorems giving sufficient conditions for a Banach space to be isomorphically polyhedral. These conditions mainly involve the notion of boundary. A subset \(B\) of the unit sphere \(S_{X^\ast}\) of a dual Banach space \(X^\ast\) is a boundary for the space \(X\) if, for every \(x \in X\), there is \(x^\ast \in B\) such that \(x^\ast (x) = \| x \|\) (see [\textit{G. Godefroy}, Math. Ann. 277, 173--184 (1987; Zbl 0597.46015)]). In their Theorem 1, it is proved that \(X\) is a separable isomorphically polyhedral Banach space whenever it satisfies one of three equivalent conditions; the first two are somewhat technical and cannot be given here, but the third is: the unit sphere \(S_X\) of \(X\) can be covered by a sequence of slices \(S (t_n, \alpha_n) = \{ x \in B_X \, ; \;t_n (x) \geq 1 - \alpha_n\}\), where \(t_n \in S_{X^\ast}\) and \(\{\alpha_n\}\) is a sequence of positive real numbers converging to \(0\).