Non-representability of finite projective planes by convex sets

TL;DR

This paper proves that finite projective planes cannot be represented by convex sets in any fixed dimension, answering a longstanding open question and revealing limitations of convex representations in geometric combinatorics.

Contribution

It establishes the non-existence of convex set representations for all finite projective planes in any fixed dimension, combining combinatorial and geometric techniques.

Findings

Finite projective planes cannot be represented by convex sets in any fixed dimension.

Existence of 2-collapsible complexes that are not d-representable for any d.

Strengthens previous results on the limitations of convex set representations.

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 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.