用莫比乌斯变换求Koebe多面体的中心
Centering Koebe polyhedra via Möbius transformations
摘要：
圆填充定理的一个变体指出，任何凸多面体的组合类都包含以原点为中心的单位球面的中间划线元素，并且这些代表在球面的莫比乌斯变换之前是唯一的。受此结果的启发，各种论文研究了莫比乌斯变换下的中心球形配置问题。特别是，Springborn证明，任何离散点集的领域有一个莫比乌斯变换映射到一个集的重心是原点，这意味着组合类的任何凸多面体包含一个元素midsribed到一个领域的额外财产，重心的切点是中心的领域。这一结果得到加强巴登，克兰和Kazhdan谁表明，同样的想法适用于任何合理的好措施定义的领域。本文的目的是表明，Springborn的声明仍然是正确的，如果我们取代重心的切点由许多其他多面体中心。证明是基于对定义在双曲空间中的某些向量场的积分曲线的拓扑性质的研究。我们还表明，大多数中心的Koebe多面体不能得到的中心的一个合适的措施定义在球面上。
英文摘要：
A variant of the Circle Packing Theorem states that the combinatorial class of any convex polyhedron contains elements midscribed to the unit sphere centered at the origin, and that these representatives are unique up to Möbius transformations of the sphere. Motivated by this result, various papers investigate the problem of centering spherical configurations under Möbius transformations. In particular, Springborn proved that for any discrete point set on the sphere there is a Möbius transformation that maps it into a set whose barycenter is the origin, which implies that the combinatorial class of any convex polyhedron contains an element midsribed to a sphere with the additional property that the barycenter of the points of tangency is the center of the sphere. This result was strengthened by Baden, Krane and Kazhdan who showed that the same idea works for any reasonably nice measure defined on the sphere. The aim of the paper is to show that Springborn's statement remains true if we replace the barycenter of the tangency points by many other polyhedron centers. The proof is based on the investigation of the topological properties of the integral curves of certain vector fields defined in hyperbolic space. We also show that most centers of Koebe polyhedra cannot be obtained as the center of a suitable measure defined on the sphere.
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