Martin Tancer recently found a very beautiful proof that finite projective planes can’t be represented by convex sets in any fixed dimension. This was asked in the paper entitled “Transversal numbers for hypergraphs arising in geometry” by Noga Alon, Gil Kalai, Jiri Matousek and Roy Meshulam some years ago. I am thankfull to Jirka Matousek for telling me about this development.
Here’s a very rough sketch of the argument: Suppose that is a finite projective plane with n points, and suppose that for every line there is a convex set in such that a subcollection of the has a common point iff the corresponding lines have a common point in the projective plane. In particular, for every we have a point in witnessing the intersection of all for which contains . (This is kind of “dual” representation compared to the usual notion, but a projective plane is self-dual so it doesn’t matter, and the proof is perhaps more intuitive in this setting.)
Look at the set . By a theorem of Janos Pach, there are disjoint subsets contained in , such that for some (enormously small) , and the convex hulls of all transversals to have a common point. Let be the corresponding subsets in the projective plane.
Now by a theorem due to Noga Alon, if is any sufficiently large subset of the projective plane, then only few lines may miss it (quantitatively, at most of the lines avoid ). Hence, since each occupies a fixed fraction of , most of the lines intersect all the simultaneously. But then all the convex sets corresponding to these lines intersect, and they shouldn’t since in the projective plane only at most lines may have a common point. QED
This result shows that a simplicial complex can fail to be the nerve of a family of convex sets in even if it is 2-collapsible. (See this post for the termonology.) This is quite a remarkable result.