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

Authors: Richard M. Aron, Robert H. Lohman and Antonio Suárez
Journal: Proc. Amer. Math. Soc. 111 (1991), 151-155
MSC: Primary 46B20; Secondary 46E30
MathSciNet review: 1030732
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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.

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Keywords: $ \lambda $-property, extreme point, convex series, Orlicz spaces, rotund Banach spaces
Article copyright: © Copyright 1991 American Mathematical Society