A variant of the Circle Packing Theorem states that the combinatorial class of any convex polyhedron contains elements, called Koebe polyhedra, 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.

中文翻译：

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通过Möbius变换使Koebe多面体居中

圆堆积定理的一个变体指出，任何凸多面体的组合类都包含被称为Koebe多面体的元素，这些元素位于以原点为中心的单位球体的中间，并且这些代表在球体的Möbius变换之前都是唯一的。受此结果的启发，各种论文都研究了在Möbius变换下定心球形结构的问题。斯普林伯恩（Springborn）特别证明，对于球体上的任何离散点集，都有一个Möbius变换将其映射到一个以重心为原点的集合中，这意味着任何凸多面体的组合类都包含一个元素，该元素与球的中点齐平相切点的重心是球体中心的附加属性。巴登加强了这一结果，克雷恩（Krane）和卡兹丹（Kazhdan）证明了相同的想法适用于该球体上定义的任何合理的度量。本文的目的是表明，如果我们用许多其他多面体中心替换相切点的重心，则Springborn的陈述仍然正确。该证明是基于对双曲空间中定义的某些矢量场的积分曲线的拓扑性质的研究。我们还表明，不能获得Koebe多面体的大多数中心作为球体上定义的合适度量的中心。该证明是基于对双曲空间中定义的某些矢量场的积分曲线的拓扑性质的研究。我们还表明，不能获得Koebe多面体的大多数中心作为球体上定义的合适度量的中心。该证明是基于对双曲空间中定义的某些矢量场的积分曲线的拓扑性质的研究。我们还表明，不能获得Koebe多面体的大多数中心作为球体上定义的合适度量的中心。