We consider the two classes of closed convex sets which are stable under dilatations and shrinkings respectively (homotheties of rates greater than one and less than one respectively). We define conjugacies for the classes of functions whose sublevel sets belong to these classes. For such functions, the conjugate can be defined on the dual space and an extra parameter is not needed. We apply these notions to the maximization of a convex function on a convex set and to the minimization of a convex function on the set of points outside a convex subset. We introduce several dual problems related to each of these problems and we give conditions
ensuring there is no duality gap.