Convexity ranks in higher dimensions

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Abstract

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 Si is the unit sphere in C(Ki) 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.

Original languageEnglish
Pages (from-to)143-163
Number of pages21
JournalFundamenta Mathematicae
Volume164
Issue number2
DOIs
StatePublished - 1 Jan 2000

Keywords

  • Cantor-Bendixson degree
  • Continuum hypothesis
  • Convexity
  • Convexity number
  • Polish vector space

ASJC Scopus subject areas

  • Algebra and Number Theory

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