Countably convex G_δ sets

We investigate countably convex G_δ 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 ⊆ S so that for every x ∈ P and open neighborhood u of x there exists a finite set X ⊆ P ∩ u such that conv(X) ⊈ S. For closed sets this condition is also necessary. We show that for countably convex G_δ 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_δ subsets of finite-dimensional spaces. The results distinguish dimension d ≤ 3 from dimension d ≥ 4: in a countably convex G_δ subset of ℝ₃ all cliques are scattered, whereas in ℝ₄ a countably convex G_δ set may contain a dense-in-itself clique.