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## Studia Mathematica

2003 | 157 | 2 | 97-120
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### On the Kunen-Shelah properties in Banach spaces

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We introduce and study the Kunen-Shelah properties $KS_{i}$, i = 0,1,...,7. Let us highlight some of our results for a Banach space X: (1) X* has a w*-nonseparable equivalent dual ball iff X has an ω-polyhedron (i.e., a bounded family $KS_{i}$, i = 0,1,...,7. Let us highlight some of our results for a Banach space X: (1) X* has a w*-nonseparable equivalent dual ball iff X has an ω-polyhedron (i.e., a bounded family ${x_{i}}_{i<ω}$ such that $x_{j} ∉ \overline{co}(x_{i}: i ∈ ω∖{j}})$ for every j ∈ ω) iff X has an uncountable bounded almost biorthogonal system (UBABS) of type η for some η ∈ [0,1) (i.e., a bounded family ${(x_{α},f_{α})}_{1≤α<ω} ⊂ X × X*$ such that $f_{α}(x_{α}) = 1$ and $|f_{α}(x_{β})| ≤ η$ if α ≠ β); (2) if X has an uncountable ω-independent system then X has an UBABS of type η for every η ∈ (0,1); (3) if X does not have the property (C) of Corson, then X has an ω-polyhedron; (4) X has no ω-polyhedron iff X has no convex right-separated ω-family (i.e., a bounded family ${x_{i}}_{i<ω}$ such that $x_{j} ∉ \overline{co}({x_{i}: j < i < ω})$ for every j ∈ ω) iff every w*-closed convex subset of X* is w*-separable iff every convex subset of X* is w*-separable iff μ(X) = 1, μ(X) being the Finet-Godefroy index of X (see [1]).
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97-120
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2003
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• Department of Mathematics, Pedagogical University, Shevchenko st. 1, Kirovograd 2506, Ukraine
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