This paper mainly concerns the topological nature of uniformly convexifiable sets in general Banach spaces: A sufficient and necessary condition for a bounded closed convex set C of a Banach space X to be uniformly convexifiable (i.e. there exists an equivalent norm on X which is uniformly convex on C) is that the set C is super-weakly compact, which is defined using a generalization of finite representability. The proofs use appropriate versions of classical theorems, such as James' finite tree theorem, Enflo's renorming technique, Grothendieck's lemma and the Davis-Figiel-Johnson-Pełczyński lemma.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm199-2-2,
author = {Lixin Cheng and Qingjin Cheng and Bo Wang and Wen Zhang},
title = {On super-weakly compact sets and uniformly convexifiable sets},
journal = {Studia Mathematica},
volume = {196},
year = {2010},
pages = {145-169},
zbl = {1252.46009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm199-2-2}
}
Lixin Cheng; Qingjin Cheng; Bo Wang; Wen Zhang. On super-weakly compact sets and uniformly convexifiable sets. Studia Mathematica, Tome 196 (2010) pp. 145-169. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm199-2-2/