Convexity numbers of closed sets in ℝⁿ

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 ℝⁿ 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) > א₀} 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 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.