This paper considers H(infinity) control problems involving discrete-time uncertain linear systems. The uncertainty is supposed to belong to convex-bounded domains, and no additional assumptions are made (as, for instance, matching conditions). Two H(infinity) guaranteed cost control problems are solved. The first one concerns the determination of a state feedback gain (if one exists) in such way the H(infinity) norm of a certain transfer function remains bounded by a prespecified H(infinity) level for all possible models. The second one includes this bound as an additional variable to be minimized to achieve the smallest feasible limiting bound. The results follow from the simple geometry of those problems which are shown to be convex in the particular parametric space under consideration. An example illustrates the theory.