Convex Union Representability and Convex Codes

We introduce and investigate $d$-convex union representable complexes: the complexes that arise as the nerve of a finite collection of convex open sets in $\mathbb R^d$ whose union is also convex. Chen, Frick, and Shiu recently proved that such complexes are collapsible and asked if all collapsible complexes are convex union representable. We disprove this by showing that there exist shellable and collapsible complexes that are not convex union representable; there also exist non-evasive complexes that are not convex union representable. In the process we establish several necessary conditions for a complex to be convex union representable such as: that such a complex $\Delta$ collapses onto the star of any face of $\Delta$, that the Alexander dual of $\Delta$ must also be collapsible, and that if $k$ facets of $\Delta$ contain all free faces of $\Delta$, then $\Delta$ is $(k-1)$-representable. We also discuss some sufficient conditions for a complex to be convex union representable. The notion of convex union representability is intimately related to the study of convex neural codes. In particular, our results provide new families of examples of non-convex neural codes.