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.

翻译：Conway 和 Guy 公布了第一个此对象存在的证据, 并随后建造了 Bezdek 和 Reshetov 。 这些示例为单数单数单数单数单数单数单数单数线( 即仅有一个稳定或不稳定的静态平衡点的属性) 的最小面孔和垂直值建立了单数的单数属性。 通过证明不存在单数单数单数单数单数单数单数单数单数单数单数单数单数单数平衡点, Conway 和 Guy 已经为同一问题建立了第一个对象的存在的证据, 其次是 Bezdek 和 Reshetov 的构造。 这些示例为单数 14, V\leq 18 和 18, 以各自的 emph{uperb{upbrbrbrb 框为单位值。 $ V. V. geq 5 边框也同样适用于每点的正数单数单数单数单数单数单数单数单数单数的正数值单数的正数值单数值正数值的正数值值值值值值值值值值值值值值值值值值值值值值值值。 。