Convexity numbers of closed sets in ℝn

Stefan Geschke, Menachem Kojman

Research output: Contribution to journalArticlepeer-review

10 Scopus citations

Abstract

For n > 2 let In be the σ-ideal in P(nω) generated by all sets which do not contain n equidIstant points in the usual metric on nω. For each n > 2 a set Sn is constructed in ℝn so that the σ-ideal which is generated by the convex subsets of Sn restricted to the convexity radical K(Sn) is isomorphic to In. Thus cov(In) is equal to the least number of convex subsets required to cover Sn - the convexity number of Sn. For every non-increasing function f : ω/2 → {κ ∈ card: cf(κ) > א0} we construct a model of set theory in which cov(In) = f(n) for each n ∈ ω/2. When f is strictly decreasing up to n, n - 1 uncountable cardinals are simultaneously realized as convexity numbers of closed subsets of ℝn. It is conjectured that n, but never more than n, different uncountable cardinals can occur simultaneously as convexity numbers of closed subsets of ℝn. This conjecture is true for n = 1 and n = 2.

Original languageEnglish
Pages (from-to)2871-2881
Number of pages11
JournalProceedings of the American Mathematical Society
Volume130
Issue number10
DOIs
StatePublished - 1 Oct 2002
Externally publishedYes

Keywords

  • Convex cover
  • Convexity number
  • Covering number
  • Forcing extension
  • n-space

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