## 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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### References:

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