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On super-weakly compact sets and uniformly convexifiable sets. (English) Zbl 1252.46009
A famous result of P. Enflo states that a Banach space is superreflexive if and only if it is uniformly convexifiable. R. C. James characterized superreflexivity in terms of the so-called finite tree property. Recall that a Banach space is said to be superreflexive whenever every Banach space finitely representable in it is reflexive. Here, the authors present a localized setting for the previous results. The main theorem reads as follows. A closed convex bounded subset of a Banach space is uniformly convexifiable if and only if it is super-weakly compact. Those notions are the natural counterpart of the corresponding global notions. More precisely, the authors introduce the following definitions. (i) A convex subset \(A\) of a Banach space is said to be uniformly convex if, for every \(x_0\in A\), the function \(x\mapsto\|x-x_0\|^2\) is uniformly convex on \(A\), and \(A\) is said to be uniformly convexifiable if there exist an equivalent norm on \(X\) such that \(A\) is uniformly convex with respect to this norm. (ii) Given two convex subsets \(A\) and \(B\) of a Banach space \(X\), \(A\) is said to be finitely representable in \(B\) if, for every \(\varepsilon>0\), each \(n\)-simplex in \(A\) can be \((1+\varepsilon)\)-affinely embedded into \(B\). (iii) A bounded closed convex subset \(C\) of \(X\) is said to be super-weakly compact if, every convex set which is finitely representable in \(C\), is relatively weakly compact.
As another example of how “global” theorems appear in this setting, the authors prove that, for a closed bounded convex subset \(C\) of a Banach space, the following statements are equivalent. (i) \(C\) is not superreflexive. (ii) \(C\) has the finite tree property. (iii) There exists \(\theta>0\) such that, for every \(n\in\mathbb N\), there are \(x_1,\ldots,x_n\) in \(C\) such that, for all \(1\leq k\leq n-1\), \(\text{dist}(\text{co}\{x_1,\ldots,x_{k}\},\text{co}\{x_{k+1},\ldots,x_n\})>\theta\).
The main ingredients in the proofs are the classical James’ finite tree theorem, Enflo’s renorming technique, Grothendieck’s lemma and the Davis-Figiel-Johnson-Pełczyński lemma.

46B20 Geometry and structure of normed linear spaces
46B03 Isomorphic theory (including renorming) of Banach spaces
46B50 Compactness in Banach (or normed) spaces
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