The \(\lambda\)-function in Banach spaces. (English) Zbl 0675.46006

Banach space theory, Proc. Res. Workshop, Iowa City/Iowa 1987, Contemp. Math. 85, 345-354 (1989).
[For the entire collection see Zbl 0669.00012.]
Let X be a Banach space, B its unit ball and ext B the set of the extreme points of B. A triple \((e,y,\lambda)\in ext B\times B\times (0,1]\) is called amenable to x if \(x=\lambda e+(1-\lambda)y\). The space X is said to have the \(\lambda\)-property if each \(x\in B\) admits an amenable triple. In this case, let \(\lambda(x)=\sup \{\lambda:(e,y,\lambda)\) is amenable to \(x\}\), \(x\in B\). These notions were defined and studied by R. M. Aron and R. H. Lohman [Pac. J. Math. 127, 209-231 (1987; Zbl 0662.46020)]. Continuing these investigations, the author shows in this paper that \(\lambda\)-property and boundedness away from 0 of the function \(\lambda\) imply that each point in B is expressible as a convex series of extreme points of B. A flatness condition, guaranteing that the function \(\lambda\) is Lipschitz on B, is also considered.
The paper ends with the following problem: If \(\lambda (x)=1\) for all \(x\in X\), with \(\| x\| =1\) is X strictly convex?
Reviewer: S.Cobzaş


46B20 Geometry and structure of normed linear spaces
46B99 Normed linear spaces and Banach spaces; Banach lattices