In this paper the Lyapunov function given by the infinity norm is considered. A new proof of the theorem of Molchanov and Pyatintskii is given. The presented proof enables us to give an explicit calculation of the matrix which was only demonstrated to exist in the original paper, Combining the obtained result with the idea of scaling faces of a polytope gives the method of fitting the shape of level surfaces, of the Lyapunov function given by the infinity norm, to the specified demands.