Zsolt Lángi: A solution to some problems of Conway and Guy on monostable polyhedra

Abstract: A convex polyhedron is called monostable if it can rest in stable position only on one of its faces. In this talk we investigate three questions of Conway, regarding monostable polyhedra, from the open problem book of Croft, Falconer and Guy (Unsolved Problems in Geometry, Springer, New York, 1991), which first appeared in the literature in a 1969 paper. In this note we answer two of these problems and conjecture an answer for 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 `polyhedral Gömböc', that is, a convex polyhedron with only one stable and one unstable point.