可内接扇 I：内接锥与虚拟多胞形
Inscribable fans I: Inscribed cones and virtual polytopes
AI总结：
本文研究内接于球面的多胞形，引入内接锥与内接虚拟多胞形，证明其封闭性及多项式时间可判定性，并分类二维可内接扇及若干多胞形类，关联反射群胚。
摘要：
我们研究内接于球面且与给定多胞形 $P$ 法向等价（或强同构）的多胞形。我们证明了相关的多胞形空间，称为 $P$ 的内接锥，在闵可夫斯基加法下封闭。内接锥被解释为理想双曲多胞形的类型锥以及 Delaunay 细分的形变空间。特别地，检验是否存在与 $P$ 法向等价的内接多胞形是多项式时间可解的。\n法向等价在法向扇的层面上决定，我们研究各类多胞形和扇（包括简单、单纯和偶类）的内接锥结构。我们在二维情形下对（虚拟）可内接扇进行了分类，并对可内接的 permutahedra 和 nestohedra 进行了分类。\n本文的第二个目标是引入内接虚拟多胞形。具有固定法向扇 $\mathcal{N}$ 的多胞形在闵可夫斯基加法下构成一个幺半群，其相关的 Grothendieck 群称为 $\mathcal{N}$ 的类型空间。类型空间的元素对应于形式闵可夫斯基差，并自然带有顶点，从而具有可内接性的概念。我们证明了内接虚拟多胞形构成一个子群，即使 $\mathcal{N}$ 没有实际的内接多胞形，该子群也可能非平凡。\n我们将内接虚拟多胞形与有向粒子轨迹联系起来，即球内粒子在受限方向下的分段线性轨迹。状态空间产生由反射生成的连通群胚，称为反射群胚。反射群胚的自同态群可视为轨迹的离散和乐群，我们确定了它们何时为反射群。
英文摘要：
We investigate polytopes inscribed into a sphere that are normally equivalent (or strongly isomorphic) to a given polytope $P$. We show that the associated space of polytopes, called the inscribed cone of $P$, is closed under Minkowski addition. Inscribed cones are interpreted as type cones of ideal hyperbolic polytopes and as deformation spaces of Delaunay subdivisions. In particular, testing if there is an inscribed polytope normally equivalent to $P$ is polynomial time solvable. Normal equivalence is decided on the level of normal fans and we study the structure of inscribed cones for various classes of polytopes and fans, including simple, simplicial, and even. We classify (virtually) inscribable fans in dimension $2$ as well as inscribable permutahedra and nestohedra. A second goal of the paper is to introduce inscribed virtual polytopes. Polytopes with a fixed normal fan $\mathcal{N}$ form a monoid with respect to Minkowski addition and the associated Grothendieck group is called the type space of $\mathcal{N}$. Elements of the type space correspond to formal Minkowski differences and are naturally equipped with vertices and hence with a notion of inscribability. We show that inscribed virtual polytopes form a subgroup, which can be non-trivial even if $\mathcal{N}$ does not have actual inscribed polytopes. We relate inscribed virtual polytopes to routed particle trajectories, that is, piecewise-linear trajectories of particles in a ball with restricted directions. The state spaces gives rise to connected groupoids generated by reflections, called reflection groupoids. The endomorphism groups of reflection groupoids can be thought of as discrete holonomy groups of the trajectories and we determine when they are reflection groups.
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