Sunflowers of Convex Open Sets

A sunflower is a collection of sets $\{U_1,\dots ,U_n\}$ such that the pairwise intersection $U_i\cap U_j$ is the same for all choices of distinct $i$ and $j$. We study sunflowers of convex open sets in ${\mathbb{R}}^d$, and provide a Helly-type theorem describing a certain “rigidity” that they possess. In particular we show that if $\{U_1,\dots ,U_{d+1}\}$ is a sunflower in $\mathbb R^d$,thenanyhyperplanethatintersectsall$U_i$mustalsointersect$\bigcap_{i=1}^{d+1} U_i$\mathrm{.}Weuseourresultstodescribeacombinatorialcode$\mathcal C_n$forall$n\ge 2$ 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 ${\mathbf{P}}_{\mathbf{C}\mathbf{o}\mathbf{d}\mathbf{e}}$.