## 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.