We study model predictive control (MPC) schemes for discrete-time linear time-invariant systems with constraints on inputs and states, that can be formulated using a linear program (LP). In particular; we focus our attention on performance criteria based on a mixed 1/infinity-norm, namely, 1-norm with respect to time and infinity-norm with respect to space. First we provide a method to compute the terminal weight so that closed-loop stability is achieved. We then show that the optimal control profile is a piecewise affine and continuous function of the initial state and briefly describe the algorithm to compute it. The piecewise affine form allows to eliminate online LP, as the computation associated with MPC becomes a simple function evaluation. Besides practical advantages, the availability of the explicit structure of the MPC controller provides an insight into the type of control action in different regions of the state space, and highlights possible conditions of degeneracies of the LP, such as multiple optima.