Ball generated property of direct sums of Banach spaces

TL;DR

This paper investigates the stability of the ball generated property (BGP) in Banach spaces under certain direct sum constructions, extending previous results to a broader class of norms.

Contribution

It proves that the BGP is preserved under direct sums of Banach spaces with respect to a wide class of smooth, normalized norms on ^2, generalizing earlier stability results.

Findings

BGP is stable under direct sums with specific smooth norms.

Extension of BGP stability beyond _0- and a_p-sums.

Applicable to a broad class of absolute, normalized, smooth norms.

Abstract

A Banach space $X$ is said to have the ball generated property (BGP) if every closed, bounded, convex subset of $X$ can be written as an intersection of finite unions of closed balls. In 2002 S. Basu proved that the BGP is stable under (infinite) $c_0$- and $\ell^p$-sums for $1<p<\infty$. We will show here that for any absolute, normalised norm $\|\cdot\|_E$ on $\mathbb{R}^2$ satisfying a certain smoothness condition the direct sum $X\oplus_E Y$ of two Banach spaces $X$ and $Y$ with respect to $\|\cdot\|_E$ enjoys the BGP whenever $X$ and $Y$ have the BGP.