Recently, a new Lyapunov function based on the convex hull of ellipsoids was introduced for the study of uncertain and/or time-varying linear discrete-time systems with/without constraints. The new Lyapunov function has many attractive features such as: i) the associated ellipsoids are not required to be robustly invariant; ii) the design conditions are formulated as linear matrix inequality constraints. The control law is obtained by solving a convex optimization problem online. This optimization problem generally does not have a closed-form solution, and hence it is solved by numerical methods. In this paper, we intend to complement the results by analyzing the geometric structures of the solution of the optimization problem, and of the control law. In particular, we show that the control law is a piecewise linear and Lipschitz continuous function of the state.