  We prove that there is no d such that all finite projective planes can be
represented by convex sets in R^d, answering a question of Alon, Kalai,
Matousek, and Meshulam. Here, if P is a projective plane with lines
l_1,...,l_n, a representation of P by convex sets in R^d is a collection of
convex sets C_1,...,C_n in R^d such that C_{i_1},...,C_{i_k} have a common
point if and only if the corresponding lines l_{i_1},...,l_{i_k} have a common
point in P. The proof combines a positive-fraction selection lemma of Pach with
a result of Alon on "expansion" of finite projective planes. As a corollary, we
show that for every $d$ there are 2-collapsible simplicial complexes that are
not d-representable, strengthening a result of Matousek and the author.
