Non-representability of finite projective planes by convex sets
Martin Tancer · Proceedings of the American Mathematical Society · 2010
We prove that there is no d d such that all finite projective planes can be represented by convex sets in R d \mathbb {R}^d , answering a question of Alon, Kalai, Matoušek, and Meshulam. Here, if P \mathbb P is a projective plane with lines ℓ 1 , … , ℓ n \ell _1,\ldots ,\ell _n , a representation of P \mathbb P by convex sets in R d \mathbb {R}^d is a collection of convex sets C 1 , … , C n ⊆ R d C_1,\ldots ,C_n \subseteq \mathbb {R}^d such that C i 1 , C i 2 , … , C i k C_{i_1},C_{i_2},\ldots ,C_{i_k} have a common point if and only if the corresponding lines ℓ i 1 , …