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

TL;DR

This paper investigates specific problems about monostable convex polyhedra, providing solutions to two questions posed by Conway and Guy, and proposing a conjecture on the third, using approximation theorems related to equilibrium points.

Contribution

It answers two longstanding questions on monostable polyhedra and introduces a new theorem linking smooth convex bodies and their equilibrium points.

Findings

Solved two of Conway and Guy's problems on monostable polyhedra.

Proved the existence of a convex polyhedron with exactly one stable and one unstable equilibrium point.

Developed a general approximation theorem for convex bodies and their equilibrium points.

Abstract

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.