Non-representability of finite projective planes by convex sets

Abstract

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 C ℝ d such that C i1 , C i2 , ... , C ik have a common point if and only if the corresponding lines l i1 , ... , l ik 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.