A note on geometrical properties of Banach spaces using ψ-direct sums

Let X be a Banach space and ψ a continuous convex function on [0, 1] satisfying certain conditions. Let X ⊕ψ X be the ψ-direct sum of X. In this note, we characterize the strict convexity, uniform convexity and uniformly non-squareness of Banach spaces using ψ-direct sums, which extends the well-known characterization of these spaces. © 2006 Elsevier Inc. All rights reserved