A set-theoretical approach based on Minkowski-Lyapunov functions is proposed to study the stability of polytopic systems in the linear parameter-varying (LPV) form. In contrast to the one based on quadratic Lyapunov functions, the proposed approach provides necessary and sufficient conditions for stability in both discrete and continuous time. Remarkably, the resulting conditions are computationally tractable and can be verified using algorithms with guaranteed convergence, in contrast to other set-theoretical tools. In addition, the proposed method is shown to be flexible enough to incorporate knowledge of the system, such as restrictions on the rate of variation of the parameters or the kind of disturbances and/or unmodeled dynamics encompassed, allowing to establish nonconservative robust stability conditions. Two illustrative examples are added to show the main advantages.