Rotund complex normed linear spaces

Abstract

We show that rotundity in a complex normed linear space is equivalent to the property that for any distinct vectors x and y of unit norm, a complex number α \alpha may be found for which ‖ α x + ( 1 − α ) y ‖ > 1 \left \| {\alpha x + (1 - \alpha )y} \right \| > 1 . This leads to a natural proof of a result due to Taylor and Foguel on the uniqueness of Hahn-Banach extensions.