Polyhedrality in pieces. (English) Zbl 1312.46014
A finite-dimensional space is called polyhedral if its unit ball is a polytope. There are several notions of polyhedrality in infinite-dimensional spaces (cf. [V. P. Fonf and L. Veselý, Can. J. Math. 56, No. 3, 472–494 (2004; Zbl 1068.46007)]) and the authors consider the original notion given by V. Klee [Acta Math. 102, 79–107 (1959; Zbl 0094.16802)]: A Banach space is said to be polyhedral when the unit balls of its finite-dimensional subspaces are polytopes. The results of the authors give conditions for replacing the norm on a given Banach space with an equivalent polyhedral norm.
The main tool used is the following property: Let \(X\) be a Banach space. We say that a set \(F \subset X^*\) has property \((*)\) if, for every \(\omega^*\)-limit point \(g\) of \(F\) (i.e., any \(\omega^*\)-neighborhood of \(g\) contains infinitely many points of \(F\)), we have \(g(x) < 1\) whenever \(\sup\{f(x): f \in F\} = 1 \). A. Gleit and R. McGuigan [Proc. Am. Math. Soc. 33, 398–404 (1972; Zbl 0244.46019)] introduced property \((*)\) in the particular case of the set of extreme points of the dual unit ball \(\text{ext}(B_{X^*})\) and showed that \(X\) is polyhedral if \(\text{ext}(B_{X^*})\) has property \((*)\).
Recall that a subset \(B \subset B_{X^*}\) is called a boundary if, for every \(x \in S_X\), there exists \(f_x \in B\) such that \(f_x(x) = 1\). Generalizing this notion, a set \(F \subset X^*\) is said to be a relative boundary if, whenever \(x \in X\) satisfies \(\sup \{ f(x): f\in F\} =1\), then there exists \(f_x \in F\) such that \(f_x(x) = 1\).
Based on a countable decomposition of the unit sphere \(S_X\), the authors find the following sufficient condition for constructing an equivalent polyhedral norm: Let \(X\) be a Banach space and suppose that we have sets \(S_n \subset S_X\) and an increasing sequence \(H_n \subset B_{X^*}\) of relative boundaries, such that \(S_X = \bigcup_{n=1}^\infty S_n\) and the numbers \[ b_n = \inf\{\sup\{h(x): h \in H_n\}: x \in S_n\} \] are strictly positive and converge to 1. Then, for a suitable sequence \((a_n)_{n =1}^\infty\), the set \[ F = \bigcup_{n=1}^\infty a_n (H_n \setminus H_{n-1}) \] is a boundary of an equivalent norm \(||| \cdot |||\). Moreover, if each \(H_n\) has property \((*)\), then \(F\) has property \((*)\) and \(||| \cdot |||\) is polyhedral.
The main tool used is the following property: Let \(X\) be a Banach space. We say that a set \(F \subset X^*\) has property \((*)\) if, for every \(\omega^*\)-limit point \(g\) of \(F\) (i.e., any \(\omega^*\)-neighborhood of \(g\) contains infinitely many points of \(F\)), we have \(g(x) < 1\) whenever \(\sup\{f(x): f \in F\} = 1 \). A. Gleit and R. McGuigan [Proc. Am. Math. Soc. 33, 398–404 (1972; Zbl 0244.46019)] introduced property \((*)\) in the particular case of the set of extreme points of the dual unit ball \(\text{ext}(B_{X^*})\) and showed that \(X\) is polyhedral if \(\text{ext}(B_{X^*})\) has property \((*)\).
Recall that a subset \(B \subset B_{X^*}\) is called a boundary if, for every \(x \in S_X\), there exists \(f_x \in B\) such that \(f_x(x) = 1\). Generalizing this notion, a set \(F \subset X^*\) is said to be a relative boundary if, whenever \(x \in X\) satisfies \(\sup \{ f(x): f\in F\} =1\), then there exists \(f_x \in F\) such that \(f_x(x) = 1\).
Based on a countable decomposition of the unit sphere \(S_X\), the authors find the following sufficient condition for constructing an equivalent polyhedral norm: Let \(X\) be a Banach space and suppose that we have sets \(S_n \subset S_X\) and an increasing sequence \(H_n \subset B_{X^*}\) of relative boundaries, such that \(S_X = \bigcup_{n=1}^\infty S_n\) and the numbers \[ b_n = \inf\{\sup\{h(x): h \in H_n\}: x \in S_n\} \] are strictly positive and converge to 1. Then, for a suitable sequence \((a_n)_{n =1}^\infty\), the set \[ F = \bigcup_{n=1}^\infty a_n (H_n \setminus H_{n-1}) \] is a boundary of an equivalent norm \(||| \cdot |||\). Moreover, if each \(H_n\) has property \((*)\), then \(F\) has property \((*)\) and \(||| \cdot |||\) is polyhedral.
Reviewer: Simon Lücking (Basel)
MSC:
| 46B03 | Isomorphic theory (including renorming) of Banach spaces |
| 46B20 | Geometry and structure of normed linear spaces |
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