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 ${\displaystyle \alpha <{\omega }_1}$ so that ϱ(x) < α for all x ∈ 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 ${\displaystyle {ℝ}^2}$ [3]. As an application of ϱ to Banach space geometry, it is proved that for every ${\displaystyle \alpha <{\omega }_1}$, 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 ${\displaystyle K_1,K_1}$ are countable compact metric spaces and ${\displaystyle S_i}$ is the unit sphere in ${\displaystyle C\left(K_i\right)}$ with the sup-norm, i = 1,2, then ${\displaystyle \varrho \left(S_1\right)=\varrho \left(S_2\right)}$ if and only if ${\displaystyle K_1}$ and ${\displaystyle K_2}$ 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 ${\displaystyle {ℝ}^2}$