Combinatorial inscribability obstructions for higher-dimensional polytopes

Joseph Doolittle, Jean-Philippe Labbé, Carsten E. M. C. Lange, Rainer Sinn, Jonathan Spreer and Günter M. Ziegler

Abstract

For 3-dimensional convex polytopes, inscribability is a classical property which 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 4-dimensional cyclic polytopes with at least 8 vertices—all of whose faces are inscribable—are not 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 polytope with f-vector (8,28,40,20) is inscribable.