Convexity ranks in higher dimensions

A subset of a vector space is called countably convex if it is a countable union of convex sets. Classification of countably convex subsets of topological vector spaces is addressed in this paper. An ordinal-valued rank function ρ is introduced to measure the complexity of local nonconvexity points in subsets of topological vector spaces. Then ρ is used to give a necessary and sufficient condition for countable convexity of closed sets. THEOREM. Suppose that S is a closed subset of a Polish linear space. Then S is countably convex if and only if there exists a < ω₁ so that ρ(χ) < α for all χ ∈ S. Classification of countably convex closed subsets of Polish linear spaces follows then easily. A similar classification (by a different rank function) was previously known for closed subset of ℝ² [3]. As an application of ρ to Banach space geometry, it is proved that for every α < ω₁, the unit sphere of C(ωα) with the sup-norm has rank α. Furthermore, a countable compact metric space K is determined by the rank of the unit sphere of C(K) with the natural sup-norm: THEOREM. If K₁, K₁ are countable compact metric spaces and S_i is the unit sphere in C(K_i) with the sup-norm, i = 1,2, then ρ(S₁) = ρ(S₂) if and only if K₁ and K₂ are homeomorphic. Uncountably convex closed sets are also studied in dimension n > 2 and are seen to be drastically more complicated than uncountably convex closed subsets of ℝ².