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COMBINATORIAL INSCRIBABILITY OBSTRUCTIONS FOR HIGHER DIMENSIONAL POLYTOPES
Author(s) -
Doolittle Joseph,
Labbé JeanPhilippe,
Lange Carsten E. M. C.,
Sinn Rainer,
Spreer Jonathan,
Ziegler Günter M.
Publication year - 2020
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/mtk.12051
Subject(s) - polytope , dual polyhedron , combinatorics , mathematics , uniform k 21 polytope , delaunay triangulation , inscribed figure , regular polygon , birkhoff polytope , geometry , convex set , convex optimization
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 eight 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 seven 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 eight vertices, that is, no polytope with f ‐vector (20,40,28,8), is inscribable.