Shellings of Unbounded Polyhedra
arXiv:2506.07241
The paper investigates shellability of the boundary complex of unbounded polyhedra by first compactifying them, then applying these results to tropical hypersurfaces, and shows that tight spans of regular subdivisions are collapsible but not generally shellable.
Abstract
The shellability of the boundary complex of an unbounded polyhedron is investigated. To this end, it is necessary to study suitable compactifications first. Results on polyhedra can then be exploited to derive a shellability result for tropical hypersurfaces. Under the hood there is a subtle interplay between the duality of polyhedral complexes and their shellability. Translated into discrete Morse theory, that interplay also gives that the tight span of a regular subdivision is collapsible, but not shellable in general.
New title; major revision. Proofs of results concerning the shellability of a one-point compactification (Cor 4, Prop 11, Cor 18 in v1) were incorrect. Instead, we now consider truncation by a half-spaces and balls as well as tropical toric compactification