A metric characterization of the Lorenz dominance order

Abstract

A metric on the space of real N-vectors RN is defined which has the property to characterise the Lorenz dominance order X < Y for X,Y E FN. The metric d is derived from the Euclidean norm U X'U , on X* [formule] where X' denotes the vector (Xi), where X = [xi) The key element in the characterization of X Y is the inequality d(X**, Y") 5 d(x*, Yi*) for every elementary permutation x of (1, ..., N), where X," = (X"),, i.e. ~c applied to X" and where a permutation is called elementary if two consecutive coordinates are interchanged.