Mono-monostatic polyhedra with uniform point masses have at least 8 vertices (2103.12769v1)

Abstract: 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.