Sunflowers of convex open sets

A sunflower is a collection of sets {U1,…,Un} such that the pairwise intersection Ui∩Uj is the same for all choices of distinct i and j. We study sunflowers of convex open sets in Rd, and provide a Helly-type theorem describing a certain “rigidity” that they possess. In particular we show that if {U1,…,Ud+1} is a sunflower in Rd, then any hyperplane that intersects all Ui must also intersect ⋂i=1d+1Ui. We use our results to describe a combinatorial code Cn for all n≥2 which is on the one hand minimally non-convex, and on the other hand has no local obstructions. Along the way we further develop the theory of morphisms of codes, and establish results on the covering relation in the poset PCode.