## Abstract

For n > 2 let I_{n} be the σ-ideal in P(n^{ω}) generated by all sets which do not contain n equidIstant points in the usual metric on n^{ω}. For each n > 2 a set S_{n} is constructed in ℝ^{n} so that the σ-ideal which is generated by the convex subsets of S_{n} restricted to the convexity radical K(S_{n}) is isomorphic to I_{n}. Thus cov(I_{n}) is equal to the least number of convex subsets required to cover S_{n} - the convexity number of S_{n}. For every non-increasing function f : ω/2 → {κ ∈ card: cf(κ) > א_{0}} we construct a model of set theory in which cov(I_{n}) = 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 ℝ^{n}. It is conjectured that n, but never more than n, different uncountable cardinals can occur simultaneously as convexity numbers of closed subsets of ℝ^{n}. This conjecture is true for n = 1 and n = 2.

Original language | English |
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Pages (from-to) | 2871-2881 |

Number of pages | 11 |

Journal | Proceedings of the American Mathematical Society |

Volume | 130 |

Issue number | 10 |

DOIs | |

State | Published - 1 Oct 2002 |

Externally published | Yes |

## Keywords

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

## ASJC Scopus subject areas

- Mathematics (all)
- Applied Mathematics

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