PreprintCombinatorial inscribability obstructions for higherdimensional polytopesJoseph Doolittle, JeanPhilippe Labbé, Carsten E. M. C. Lange, Rainer Sinn, Jonathan Spreer and Günter M. ZieglerAbstractFor 3dimensional convex polytopes, inscribability is a classical property which is relatively wellunderstood 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 fvector of 3polytopes, there exists an inscribable polytope with that fvector. For higherdimensional polytopes, much less is known. Of course, for any inscribable polytope, all of its lowerdimensional faces need to be inscribable, but this condition does not appear to be very strong. We observe nontrivial 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 4dimensional 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 ddimensional cyclic polytopes with at least d+4 vertices are not circumscribable, and that no polytope with fvector (8,28,40,20) is inscribable. This paper is available as a pdf (528kB) file.
