# Convex Union Representability and Convex Codes

@article{Jeffs2019ConvexUR, title={Convex Union Representability and Convex Codes}, author={R. Amzi Jeffs and Isabella Novik}, journal={International Mathematics Research Notices}, year={2019} }

We introduce and investigate $d$-convex union representable complexes: the simplicial 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…

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