A characterization of B-convexity and J-convexity of Banach spaces

Abstract

In [K.-I. Mitani and K.-S. Saito, J. Math. Anal. Appl. 327 (2007), 898-907] we characterized the strict convexity, uniform convexity and uniform non-squareness of Banach spaces using $\psi$-direct sums of two Banach spaces, where $\psi$ is a continuous convex function with some appropriate conditions on $[0,1]$. In this paper, we characterize the $B_n$-convexity and $J_n$-convexity of Banach spaces using $\psi$-direct sums of $n$ Banach spaces, where $\psi$ is a continuous convex function with some appropriate conditions on a certain convex subset of $\mathbb R^n$.