This is an announcement for the paper "Ball generated property of direct sums of Banach spaces" by Jan-David Hardtke.
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.
Archive classification: math.FA
Mathematics Subject Classification: 46B20
Remarks: 9 pages
Submitted from: hardtke@math.fu-berlin.de
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1502.06224
or