Sets in a euclidean space which are not a countable union of convex subsets

We prove that if a closed planar set S is not a countable union of convex subsets, then exactly one of the following holds: (a) There is a perfect subset P⊆S such that for every pair of distinct points x, yεP, the convex closure of x, y is not contained in S. (b) (a) does not hold and there is a perfect subset P⊆S such that for every pair of points x, yεP the convex closure of {x, y} is contained in S, but for every triple of distinct points x, y, zεP the convex closure of {x, y, z} is not contained in S. We show that an analogous theorem is impossible for dimension greater than 2. We give an example of a compact planar set with countable degree of visual independence which is not a countable union of convex subsets, and give a combinatorial criterion for a closed set in R ^d not to be a countable union of convex sets. We also prove a conjecture of G. Kalai, namely, that a closed planar set with the property that each of its visually independent subsets has at most one accumulation point, is a countable union of convex sets. We also give examples of sets which possess a (small) finite degree of visual independence which are not a countable union of convex subsets.