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1 Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

1 2000

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Convexity induced by a set quantity

Falcon S., Sadarangani K.B.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2000

[Z] 40

57-67

In the geometry of Banach spaces the notion of convexity plays a very significant role and is frequently used in many branches of functional analysis. In the last few years, there have appeared some papers containing generalizations of the concept of convexity using the notion of a measure of noncompactness. The aim of this paper is to generalize the notion of convexity using the notion of a set quantity wich has been considered in. It is worthwhile mentioning that several results obtained in the geometry involving compactness conditions have counterparts in the geometry induced by a set quantity. Particularly, we introduce a modulus related to a set quantity and we obtain a generalization of a result due to Rolewicz. Moreover, we calculate this modulus for the De Blasi measures of weak noncompactness in some classics Banach spaces such as C_0, l^1, L^1 and the James space J.