Combinatorial Inscribability Obstructions for Higher-Dimensional Polytopes

TL;DR

This paper introduces new combinatorial obstructions to inscribability in higher-dimensional polytopes, demonstrating that many known classes of polytopes are not inscribable, unlike the well-understood 3D case.

Contribution

It identifies novel combinatorial obstructions to inscribability in higher dimensions and characterizes inscribability for duals of cyclic and neighborly polytopes.

Findings

Duals of 4D cyclic polytopes with ≥8 vertices are not inscribable.

Duals of cyclic 4-polytopes with ≤7 vertices are inscribable.

Certain face lattice subposets serve as forbidden structures for inscribability.

Abstract

For $3$-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 $f$-vector of $3$-polytopes, there exists an inscribable polytope with that $f$-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 $4$-dimensional cyclic polytopes with at least $8$ 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 $4$-polytopes with up to $7$ vertices are, in fact, inscribable. Moreover, we interpret this obstruction combinatorially as a forbidden subposet of the face lattice of a polytope, show that $d$-dimensional cyclic polytopes with at least $d+4$ vertices are not circumscribable, and that no dual of a neighborly $4$-polytope with $8$ vertices, that is, no polytope with $f$-vector $(20,40,28,8)$, is inscribable.