Complex strict convexity of absolute norms on Cn and direct sums of Banach spaces

We show that an absolute normalized norm on C2 is complex strictly convex if and only if the associated convex function ψ on [0, 1] satisfies ψ(t)>max{1−t, t} for all 0 < t <1. Using this result, we characterize when the direct sum X⊕ψ Y of the complex Banach spaces X and Y is complex strictly convex and complex uniformly convex. A characterization of complex strict convexity of Cn forn>2, equipped with an absolute normalized norm, is also given. © 2005 Elsevier Inc. All rights reserved