Rotundity, the C.S.R.P., and the 𝜆-property in Banach spaces

Abstract

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