Penalty functions based upon a general class of restricted dissimilarity functions

•General results about the class of restricted dissimilarity functions.•A subclass of restricted dissimilarity functions based on automorphisms.•A model of generation of penalty functions by means of automorphisms.•A characterization theorem of the generators of penalty functions of any dimension.•The existence of not convex generators is shown. In this paper the notion of restricted dissimilarity function is discussed and some general results are shown. The relation between the concepts of restricted dissimilarity function and penalty function is presented. A specific model of construction of penalty functions by means of a wide class of restricted dissimilarity functions based upon automorphisms of the unit interval is studied. A characterization theorem of the automorphisms which give rise to two-dimensional penalty functions is proposed. A generalization of the previous theorem to any dimension n > 2 is also provided. Finally, a not convex example of generator of penalty functions of arbitrary dimension is illustrated.