In 1976 Cheney and Franchetti derived sufficient and necessary (assuming the subspace is “smooth”) conditions for finite rank \(L^ 1\) projections to be minimal. One of the conditions is an equation (constancy of the Lebesgue function) and the remaining conditions are of an inequality nature. As an application of the sufficiency of these conditions, they obtained the minimal projection from \(L^ 1[-1,1]\) onto the lines \([1,t]\). In the reviewed paper a set of easily applicable necessary and sufficient equations, is developped, with no assymptions on the subspace. It is demonstrated that these equations imply certain linear equations known to be necessary in this context. It is shown that these equations admit a solution in terms of a family of best approximation problems in \(L^ 1\) and the orthogonality conditions, a solution of which has a simple geometric interpretation. Criteria for both uniqueness and nonuniqueness are obtained. As applications, authors rederive the Cheney-Franchetti example of the minimal projection onto \([1,t]\), and also obtain the minimal projection onto the quadratics \([1,t,t^ 2]\). A new proof of minimality in the setting of general compact abelian groups is given. The equations apply equally well to operators of minimal norm extending any fixed action on the given subspace and thus may be viewed as an extension of the Hahn-Banach theorem to higher dimensions in the \(L^ 1\) setting. Examples of this are also given.