NORMAL STRUCTURE AND SMOOTHNESS IN BANACH SPACES

Abstract

We proved that a Banach space X with PX(e)<e/2 for some 0 < e < infty is uniform nonsquare; and a Banach space X with PX(e)<e/2 for some 0 <e leq 1 or PX(e)<e/2-1/2 some 1 < e < infty has uniform normal structure.