Sunflowers of Convex Open Sets

TL;DR

This paper investigates sunflowers of convex open sets in Euclidean space, establishing a Helly-type theorem that reveals a rigidity property, and explores implications for combinatorial codes and morphisms.

Contribution

It introduces a Helly-type theorem for convex open set sunflowers, characterizes a minimally non-convex combinatorial code, and advances the theory of code morphisms and poset relations.

Findings

Helly-type theorem for convex open set sunflowers in d

Characterization of a minimally non-convex code _n

Development of the theory of morphisms of codes

Abstract

A sunflower is a collection of sets $\{U_1,\ldots, U_n\}$ 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 $\{U_1,\ldots, U_{d+1}\}$ is a sunflower in $\mathbb R^d$, then any hyperplane that intersects all $U_i$ must also intersect $\bigcap_{i=1}^{d+1} U_i$. We use our results to describe a combinatorial code $\mathcal C_n$ for all $n\ge 2$ 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 $\mathbf P_{\mathbf{Code}}$.