Banach spaces where convex combinations of relatively weakly open subsets of the unit ball are relatively weakly open
Volume 250 / 2020
Abstract
Ghoussoub, Godefroy, Maurey, and Schachermayer showed that in the positive face of the unit ball of $L_1[0,1]$, finite convex combinations of relatively weakly open subsets are relatively weakly open. We study this phenomenon in the closed unit balls of Banach spaces and call it property \textit{CWO}. We introduce a geometric property, called~(\textit{CO}), and show that if a finite-dimensional normed space $X$ has property~(\textit{CO}), then for any scattered locally compact Hausdorff space $K$, the space $C_0(K,X)$ has property~\textit{CWO}. Several finite-dimensional spaces are shown to have property (\textit{CO}). We present an example of a three-dimensional real Banach space for which $C_0(K,X)$ fails property \textit{CWO}. We also obtain stability results for the properties \textit{CWO} and (\textit{CO}): for instance, if a Banach space contains a complemented subspace isomorphic to $\ell _1$, then it does not have property~\textit{CWO}.
