On super-weakly compact sets and uniformly convexifiable sets
Volume 199 / 2010
Studia Mathematica 199 (2010), 145-169
MSC: Primary 46B20, 46B03, 46B50.
DOI: 10.4064/sm199-2-2
Abstract
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.
