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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.

##### MSC:
 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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