Non-representability of finite projective planes by convex sets

Martin Tancer

Abstract

It is well known that a finite projective plane cannot be represented by lines in R^d. We prove a similar result when we consider a representation by convex sets (for projective planes that arise from a finite field).

More precisely, we show that for every positive integer d there is a positive integer q_0 with the following property. Let us have a projective plane which arises over GF(q) for q \geq q_0 with the set of lines L. Then there are no convex sets C_l in d-space for lines l from L such that for every l_1,...,l_k the sets C_{l_1},...,C_{l_k} intersect if and only if the lines l_1,...,l_k meet in a point of the projective plane.

If I have enough time I will also explain the main motivation of this problem: a simplicial complex is d-representable if it is the nerve of a collection of convex sets in R^d. A simplicial complex is d-collapsible if it can be reduced to an empty complex by