Ball generated property of direct sums of Banach spaces

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.