Convexity numbers of closed sets in ℝⁿ

Abstract

For n > 2 n>2 let I n \mathcal I_n be the σ \sigma -ideal in P ( n ω ) \mathcal P(n^\omega ) generated by all sets which do not contain n n equidistant points in the usual metric on n ω n^\omega . For each n > 2 n>2 a set S n S_n is constructed in R n \mathbb {R}^n so that the σ \sigma -ideal which is generated by the convex subsets of S n S_n restricted to the convexity radical K ( S n ) K(S_n) is isomorphic to I n \mathcal I_n . Thus cov ⁡ ( I n ) \operatorname {cov}(\mathcal I_n) is equal to the least number of convex subsets required to cover S n S_n — the convexity number of S n S_n . For every non-increasing function f : ω ∖ 2 → { κ ∈ card : cf ⁡ ( κ ) > ℵ 0 } f:\omega \setminus 2\to \{\kappa \in \operatorname {card}:\operatorname {cf}(\kappa )>\aleph _0\} we construct a model of set theory in which cov ⁡ ( I n ) = f ( n ) \operatorname {cov}(\mathcal I_n)=f(n) for each n ∈ ω ∖ 2 n\in \omega \setminus 2 . When f f is strictly decreasing up to n n , n − 1 n-1 uncountable cardinals are simultaneously realized as convexity numbers of closed subsets of R n \mathbb {R}^n . It is conjectured that n n , but never more than n n , different uncountable cardinals can occur simultaneously as convexity numbers of closed subsets of R n \mathbb {R}^n . This conjecture is true for n = 1 n=1 and n = 2 n=2 .