具有均匀点质量的单单站多面体至少有8个顶点
Mono-monostatic polyhedra with uniform point masses have at least 8 vertices
摘要：
凸多面体的单站性质（即只有一个稳定或不稳定的静态平衡点的性质）一直是研究的焦点，因为康威和盖伊发表了第一个这样的对象存在的证明，其次是Bezdek和Reshetov的结构。这些示例将$F\leq 14, V\leq 18$建立为用于均匀单稳定多面体的最小数目的面和顶点的相应\emph{upper bounds}。Conway和Guy通过证明不存在单稳定的齐次四面体，建立了与$F, V \geq 5$相同的面数和顶点数的下界，并对单不稳定的情形建立了相同的下界。同样清楚的是，$F,V \geq 5$边界也适用于凸的、在每个点处具有单位质量的齐次点集（也称为多面体0-骨架），并且它们也适用于具有恰好一个稳定的和一个不稳定的平衡点（齐次骨架和0-骨架）的单-单静态多面体。在这里，我们提出了一个算法，我们提高了下界$V\geq 8$顶点（意味着$f \geq 6$面）单不稳定和单单稳定0-骨架。我们的算法似乎不太适合计算单稳定性的下界。我们指出这些困难与工作的道森和Finbow谁探讨了单稳态属性的简单在更高的层面。
英文摘要：
The monostatic property of convex polyhedra (i.e. the property of having just one stable or unstable static equilibrium point) has been in the focus of research ever since Conway and Guy published the proof of the existence of the first such object, followed by the constructions of Bezdek and Reshetov. These examples establish $F\leq 14, V\leq 18$ as the respective \emph{upper bounds} for the minimal number of faces and vertices for a homogeneous mono-stable polyhedron. By proving that no mono-stable homogeneous tetrahedron existed, Conway and Guy established for the same problem the lower bounds for the number of faces and vertices as $F, V \geq 5$ and the same lower bounds were also established for the mono-unstable case. It is also clear that the $F,V \geq 5$ bounds also apply for convex, homogeneous point sets with unit masses at each point (also called polyhedral 0-skeletons) and they are also valid for mono-monostatic polyhedra with exactly on stable and one unstable equilibrium point (both homogeneous and 0-skeletons). Here we present an algorithm by which we improve the lower bound to $V\geq 8$ vertices (implying $f \geq 6$ faces) on mono-unstable and mono-monostable 0-skeletons. Our algorithm appears to be less well suited to compute the lower bounds for mono-stability. We point out these difficulties in connection with the work of Dawson and Finbow who explored the monostatic property of simplices in higher dimensions.
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