  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.
