Rotundity, the C.S.R.P., and the $\lambda$-property in Banach spaces

Abstract:

Two open questions stemming from the $\lambda$-property in Banach spaces are solved. The following are shown to be equivalent in a Banach space $X$: (a) $X$ has the $\lambda$-property; (b) every vector in the closed unit ball of $X$ is expressible as a convex series of extreme points of the unit ball of $X$. Also, by exhibiting a class of nonrotund Orlicz spaces for which the $\lambda$-function is identically 1 on the unit spheres, we answer negatively the question of whether the $\lambda$-function characterizes rotund Banach spaces.