Complex Strongly Extreme Points in Quasi-Normed Spaces*

We study the complex strongly extreme points of bounded.subsets of continuously quasi-normed vector spaces X over C. When X is a complex normed linear space, these points are the complex analogues of the familiar real. strongly extreme points. We show that if X is a complex Banach space then the complex strongly extreme points of BX admit several equivalent formulations some of which are in terms of ‘‘pointwise’’ versions of well known moduli of complex convexity. We use this result to obtain a characterization of the complex extreme points of Blp Xj.jgI and BLp m, X . where 0-p-`, X and each Xj , jgI, are complex Banach spaces.