QUESTIONS RESULTING FROM CONVEXITY

Question 1. a) Let n > 2. Is |σ_{conv}(ℝ^{n})| = n consistent? b) Let V be an infinite dimensional Hilbert space. Can |σ_{conv}(V)| be infinite? Question 2. Let n > 2. Is |σ_{conv}(ℝ^{n})| finite? Is |σ_{conv}(ℝ^{n})| ≤ n? Question 3. Is it true that every closed S ⊆ R^{d} with no perfect clique can consistently have γ(S) < 2^{ℵ_{0}}? (true for d = 2) Question 4. Is there a closed S ⊆ ℝ^{2} whose convexity ideal is I(c_{max})? Question 5. Convexity ideals of Borel sets. In particular, G_{𝔡} sets. Question 6. Is there a nice DST classification of continuous colorings? Is the division between c_{min} and c_{max} absolute? Question 7. Is ℵ_{1} < 𝔥𝔪(c_{min}) < 2^{ℵ_{0}} consistent? Question 8. Is I(c_{max}) the smallest meager ideal? Is there a larger cardinal invariant than 𝔥𝔪(c_{max}) except 𝔠?

Relevant papers can be downloaded from:

  http://www.math.bgu.ac.il/∼kojman

Date: September 2002.