Diameter two properties and polyhedrality. (English) Zbl 1416.46013
Let \(X\) be a Banach space and \(B_X\) its closed unit ball. Recall that a Banach space \(X\) has the strong diameter two property (SD2P) if every convex combination of slices of \(B_X\) has diameter two. Note that a Banach space with the SD2P must be infinite-dimensional.
In [V. P. Fonf and L. Veselý, Can. J. Math. 56, No. 3, 472–494 (2004; Zbl 1068.46007)], various generalizations of the notion of a polytope to infinite-dimensional Banach spaces were introduced and they were called (I)-polyhedrality, (II)-polyhedrality, etc., up to (VIII)-polyhedrality. It is known that each of the previous polyhedrality notions implies the next one and none of the reverse implications holds.
The paper under review starts by observing that every infinite-dimensional (I)-polyhedral space has the SD2P, because they are non-reflexive M-embedded spaces. The main result of the paper under review says: for every \(\varepsilon>0\), there is a (II)-polyhedral Banach space \(X\) such that \(B_X\) contains slices whose diameter is smaller than \(\varepsilon\). Hence already the (II)-polyhedral Banach spaces are dramatically different from (I)-polyhedral spaces and lack the SD2P.
In [V. P. Fonf and L. Veselý, Can. J. Math. 56, No. 3, 472–494 (2004; Zbl 1068.46007)], various generalizations of the notion of a polytope to infinite-dimensional Banach spaces were introduced and they were called (I)-polyhedrality, (II)-polyhedrality, etc., up to (VIII)-polyhedrality. It is known that each of the previous polyhedrality notions implies the next one and none of the reverse implications holds.
The paper under review starts by observing that every infinite-dimensional (I)-polyhedral space has the SD2P, because they are non-reflexive M-embedded spaces. The main result of the paper under review says: for every \(\varepsilon>0\), there is a (II)-polyhedral Banach space \(X\) such that \(B_X\) contains slices whose diameter is smaller than \(\varepsilon\). Hence already the (II)-polyhedral Banach spaces are dramatically different from (I)-polyhedral spaces and lack the SD2P.
Reviewer: Johann Langemets (Tartu)
Citations:
Zbl 1068.46007References:
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