The problem of designing an internal stabilizing controller \(K\) which stabilizes \(G\) and \(G+\Delta\) for all \(\Delta\)'s with \(\|\Delta\|< r\) is a deeply studied problem, and it is well known how to design \(K\) and what is the largest radius \(r\). The authors show that there is a worst case direction in the maximal ball of uncertainties, i.e., for a perturbation lying on the boundary of this ball in this direction the system will be destabilized by all optimal controllers. This implies that it is impossible to improve the robust stability in this direction. Keeping the perturbation fixed in this worst direction, the authors solve the problem of maximizing the allowable set of perturbations. Naturally, this implies that this set becomes an ellipsoid. In the last section they use their result to obtain a new upper bound for the structured singular values.