Department of Mathematics
Convex Geometry, Continuous Colorings and Metamathematics
Stefan Geschke
Freie Universitat Berlin
I will give an overview over a number of results obtained by Kojman, Kubis, Schipperus and myself concerning certain cardinal invariants arising in convex geometry. For a subset S of a real vector space we consider the convexity number γ(S), the least cardinality of a family F of convex subsets of S which covers S. We are mainly interested in uncountable convexity numbers of closed subsets of Rn. In R1 the situation is simple. For every closed subset S of R1 either γ(S) is countable or there is a nonempty perfect subset P of S such that every convex subset of S intersects P in at most 2 points. In the latter case γ(S) = |R|.
The situation is more complicated in R2. For every closed subset S of R2 exactly one of the following two statements holds:
(1) There is a nonempty perfect subset P of S such that every convex subset of S intersects P in at most 3 points (and hence γ(S) = |R|).
(2) There is a forcing extension of the set-theoretic universe in which γ(S) < |R| (and hence there is no set P as in (1)).
The convexity numbers of closed sets satisfying (2) turn out to have a combinatorial characterization as so-called homogeneity numbers of continuous pair colorings on the Baire space N N. The metamathematical issues involved in statement (2) will be discussed briefly. The dichotomy for closed subsets of R2 cannot be generalized to higher dimensions. I will mention some results that are provable in higher dimensions.
All interested persons are welcome.