Before revision:  We prove upper and lower bounds on the ratio between the radii of an optimal but unstable solution and the radii of a topologically stable solution---the topological stability ratio---considering various metrics and various optimization criteria.
After revision:  We prove upper and lower bounds on the ratio between the radii of an optimal but unstable solution and the radii of a topologically stable solution -- the topological stability ratio -- considering various metrics and various optimization criteria.