This paper explores the problem of maximizing the size of a region of asymptotic stability (RAS) over a class of Lyapunov functions (LF) in order to estimate the domain of attraction of a continuous or discontinuous (relay) system. Analytic and numerical evidence shows that a subclass of LF may exist with multiple-tangency for the RAS problem, and can cause some serious convergence problems. Several systems are considered with two classes of LF for continuous and one class for relay systems.