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.

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高维多面体的组合可记性障碍

对于 $3$ 维凸多面体，可刻写性是一个经典属性，由于其与平面和双曲几何的 Delaunay 细分的关系而相对容易理解。特别是，可以在多项式时间内测试可写性，并且对于 $3$-polytope 的每个 $f$-vector，存在一个具有该 $f$-vector 的可写多胞体。对于更高维的多胞体，我们所知甚少。当然，对于任何可刻录的多胞体来说，它的所有低维面都需要是可刻录的，但这个条件似乎并不强。我们观察到在强加某个可刻写的面被刻录时出现的对多胞体可刻录性的非平凡的新障碍。利用这个障碍，我们证明了具有至少 8 美元顶点的 4 美元维循环多胞体的对偶——所有面都是可写的——是不可写的。这个结果在以下意义上是最优的：我们证明了具有高达 $7$ 顶点的循环 $4$-polytopes 的对偶实际上是不可写的。此外，我们将这种障碍组合地解释为多胞体面格的禁止子集，表明具有至少 $d+4$ 个顶点的 $d$ 维循环多胞体是不可外接的，并且没有邻位 $4$ 的对偶-polytope with $8$ 顶点，即没有 $f$-vector $(20,40,28,8)$ 的polytope 是不可写的。