This paper considers the synthesis of polyhedral Lyapunov functions for continuous-time dynamical systems. A proper conic partition of the state-space is employed to construct a finite set of linear inequalities in the elements of the Lyapunov weight matrix. For dynamics described by linear and polytopic differential inclusions, it is proven that the feasibility of the derived set of linear inequalities is necessary and sufficient for the existence of an infinity norm Lyapunov function. Furthermore, it is shown that the developed solution naturally applies to relevant classes of continuous-time nonlinear systems. An extension to non-symmetric polyhedral Lyapunov functions is also presented.