## 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

### MSC:

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

### Citations:

Zbl 0669.00012; Zbl 0662.46020