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Reflexivity of a Banach space with a uniformly normal structure. (English) Zbl 0548.46014

A Banach space X is said to have a uniformly normal structure if there exists a number h, \(0<h<1\), such that if C is a closed convex bounded subset of X, then there exists x in C such that \(\sup\{\| x- y\|;\quad y\in C\}\leq h\delta (C),\) where \(\delta\) (C) denotes the diameter of the set C. In this note we prove that any Banach space with a uniformly normal structure is reflexive.

MSC:

46B20 Geometry and structure of normed linear spaces
46B10 Duality and reflexivity in normed linear and Banach spaces
47H10 Fixed-point theorems
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