The convex hull of ellipsoids was suggested in the literature for robust invariance and robust controlled invariance of a constrained uncertain and/or time-varying linear discrete-time system. It was show that this class of sets can reduce significantly the conservativeness of the ellipsoidal set. However, the design conditions are  given in terms of bilinear matrix inequalities (BMIs), which are non-convex. The main purpose of this paper is to present a way to overcome this weakness by providing new convex linear matrix inequality (LMI) design conditions. It is shown that the conditions are losslessly extended to stabilization, providing then an LMI solution to the non-linear state feedback control design problem of constrained uncertain time-varying linear discrete-time systems Two examples are included with comparison to earlier solutions from the literature to illustrate the results.