Recently Cusick [4] presented a very elegant and short proof of the fact that a pair of fundamental units of a totally real cubic or quartic number field can be found by taking ‘successive minima’ of the function tr(ε2), where ε runs through the group of units and tr denotes the absolute trace. Hidden in Cusick's proof there is a general theorem, which shows how strictly convex functions can be used to find lattice-vectors extensible to a basis of a given geometrical lattice, and which we state and prove in § 3. A result analogous to Cusick's for some families of functions related to tr (ε2) is given in Theorem 1, thereby improving results of Brunotte and Halter-Koch [2], [5]. For a survey of unit groups of rank 2 and more literature the reader is also referred to [2].