Recently Deutsch, Li and Swetits [2] have studied, in Hilbert space, a dual problem (Qm) to the primal problem (P) of minimization of a special class of convex functionsfover the intersection ofmclosed convex sets, wheremis finite. In the first part of this paper we obtain, in a locally convex space, some results on problem (Qm) and on its relations with the usual Lagrangian dual problem (Q) to (P) (studied in [9]), in the case when (P) has a solution. In the second part we give some applications to duality for the distance to the intersection ofmclosed convex sets in a normed linear space, in the case when a nearest point exists. Most of our results seem to be new even in the particular cases studied in [9] (the casem= 1), [l] (duality formulas for the distance to the intersection ofmclosed half-spaces in a normed linear space) and [2].