We use a class of locally Lipschitz continuous Lyapunov functions to establish stability for a class of differential inclusions where the set-valued map on the right-hand-side comprises the convex hull of a finite number of vector fields. Starting with a finite family of continuously differentiable positive definite functions, we study conditions under which a function obtained by max-min combinations over this family of functions is a Lyapunov function for the system under consideration. For the case of linear systems, using the S-Procedure, our conditions result in bilinear matrix inequalities. The proposed construction also provides nonconvex Lyapunov functions, which are shown to be useful for systems with state-dependent switching that do not admit a convex Lyapunov function.