Countably convex $G_{\delta }$ sets
Volume 168 / 2001
Abstract
We investigate countably convex $G_{\delta }$ subsets of Banach spaces. A subset of a linear space is countably convex if it can be represented as a countable union of convex sets. A known sufficient condition for countable convexity of an arbitrary subset of a separable normed space is that it does not contain a semi-clique [9]. A semi-clique in a set $S$ is a subset $P\subseteq S$ so that for every $x\in P$ and open neighborhood $u$ of $x$ there exists a finite set $X\subseteq P\cap u$ such that $\mathop {\rm conv}(X)\not \subseteq S$. For closed sets this condition is also necessary.
We show that for countably convex $G_{\delta }$ subsets of infinite-dimensional Banach spaces there are no necessary limitations on cliques and semi-cliques.
Various necessary conditions on cliques and semi-cliques are obtained for countably convex $G_{\delta }$ subsets of finite-dimensional spaces. The results distinguish dimension $d\le 3$ from dimension $d\ge 4$: in a countably convex $G_{\delta }$ subset of ${\mathbb R}^{3}$ all cliques are scattered, whereas in ${\mathbb R}^4$ a countably convex $G_{\delta }$ set may contain a dense-in-itself clique.
