Non-representability of finite projective planes by convex sets

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.