Combinatorial Inscribability Obstructions for Higher-Dimensional Polytopes
arXiv:1910.05241
Abstract
For -dimensional convex polytopes, inscribability is a classical property that is relatively well-understood due to its relation with Delaunay subdivisions of the plane and hyperbolic geometry. In particular, inscribability can be tested in polynomial time, and for every -vector of -polytopes, there exists an inscribable polytope with that -vector. For higher-dimensional polytopes, much less is known. Of course, for any inscribable polytope, all of its lower-dimensional faces need to be inscribable, but this condition does not appear to be very strong. We observe non-trivial new obstructions to the inscribability of polytopes that arise when imposing that a certain inscribable face be inscribed. Using this obstruction, we show that the duals of the -dimensional cyclic polytopes with at least vertices---all of whose faces are inscribable---are not inscribable. This result is optimal in the following sense: We prove that the duals of the cyclic -polytopes with up to vertices are, in fact, inscribable. Moreover, we interpret this obstruction combinatorially as a forbidden subposet of the face lattice of a polytope, show that -dimensional cyclic polytopes with at least vertices are not circumscribable, and that no dual of a neighborly -polytope with vertices, that is, no polytope with -vector , is inscribable.
27 pages, 10 Figures, 4 Tables, 2 Appendices