2019/10/11 by Joseph Doolittle, Doolittle, Joseph, Jean‐Philippe Labbé +9
Computer Science · Mathematics · #52B11 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1910.05241
openalex publication_date 2019/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For 3-dimensional convex polytopes, inscribability is a classical property\nthat is relatively well-understood due to its relation with Delaunay\nsubdivisions of the plane and hyperbolic geometry. In particular,\ninscribability can be tested in polynomial time, and for every f-vector of\n3-polytopes, there exists an inscribable polytope with that f-vector. For\nhigher-dimensional polytopes, much less is known. Of course, for any\ninscribable polytope, all of its lower-dimensional faces need to be\ninscribable, but this condition does not appear to be very strong.\n We observe non-trivial new obstructions to the inscribability of polytopes\nthat arise when imposing that a certain inscribable face be inscribed. Using\nthis obstruction, we show that the duals of the 4-dimensional cyclic\npolytopes with at least 8 vertices---all of whose faces are inscribable---are\nnot inscribable. This result is optimal in the following sense: We prove that\nthe duals of the cyclic 4-polytopes with up to 7 vertices are, in fact,\ninscribable.\n Moreover, we interpret this obstruction combinatorially as a forbidden\nsubposet of the face lattice of a polytope, show that d-dimensional cyclic\npolytopes with at least d+4 vertices are not circumscribable, and that no\ndual of a neighborly 4-polytope with 8 vertices, that is, no polytope with\nf-vector (20,40,28,8), is inscribable.\n