2020/08/05 by Zsolt Lángi, Lángi, Zsolt
Mathematics · #52B10 #52B15 #70C20 #Advanced Differential Equations and Dynamical Systems #Advanced Topology and Set Theory #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2008.02090
openalex publication_date 2020/08/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
A convex polyhedron is called monostable if it can rest in stable position only on one of its faces. The aim of this paper is to investigate three questions of Conway, regarding monostable polyhedra, which first appeared in a 1969 paper of Goldberg and Guy (M. Goldberg and R.K. Guy, Stability of polyhedra (J.H. Conway and R.K. Guy), SIAM Rev. 11 (1969), 78-82). In this note we answer two of these problems and make a conjecture about the third one. The main tool of our proof is a general theorem describing approximations of smooth convex bodies by convex polyhedra in terms of their static equilibrium points. As another application of this theorem, we prove the existence of a convex polyhedron with only one stable and one unstable point.