A SOLUTION TO SOME PROBLEMS OF CONWAY AND GUY ON MONOSTABLE POLYHEDRA

The paper proves that for any n ≥ 3 and ε > 0 there exists a homogeneous monostable polyhedron with n-fold rotational symmetry arbitrarily close (in Hausdorff distance) to the unit ball, via a general approximation theorem (Theorem 2) that preserves the number and types of equilibrium points under G-invariant polyhedral truncations, and a specific construction of a smooth body K(ε) with one stable and n+1 unstable equilibria used in Theorem 1 . By contrast, the model’s Step 1 takes r(z)^2 = 1 − z^2 + ξ z; but this defines a translated sphere. After recentering at the center of mass (as the model explicitly does), one obtains an exact sphere, for which the boundary is completely degenerate with respect to equilibria, so the claim of “only two axial equilibria and no off-axis equilibria” fails. Subsequent steps rely critically on that false property, and also contradict the cited statement that no body of revolution can be monostable (attributed to Conway) . Hence the model’s argument is untenable, while the paper’s is coherent and complete.