Complex convexity and the geometry of Banach spaces

Abstract

The notion of PL-convexity was introduced in [4]. In the present article several results are proved which related PL-convexity to various aspects of the geometry of Banach spaces. The first section introduces the moduli of comples convexity and makes a comparison with the more familiar modulus of uniform convexity. It is shown that unconditional convergence of implies convergence of . In the next section the moduli and are shown to be related. The method of proof gives rise to a theorem about strict c-convexity of Lp(X) and a result on the representability in Lp(X).