MAJORIZATION AND SCHUR CONVEXITY WITH RESPECT TO PARTIAL ORDERS

Majorization and Schur convexity constitute an important tool for esta blishing inequalities, in particular, due to the Schur-Ostrowski Theor em which provides a simple characterization of Schur convex functions through local two-coordinate conditions. However, the usefulness of th e approach is limited by two stringent requirements. First, every Schu r convex function must be symmetric. Second, the necessary and suffici ent conditions for Schur convexity involve every pair of coordinates o n its (symmetric) domain. Several attempts have been made to relax the se two requirements. In particular, a more general concept of ''majori zation with respect to partial orders'' was introduced to prepare for such tasks, but it did not capture the classic theory for symmetric fu nctions. In the current paper we obtain the desired generalization by developing a theory of majorization and Schur convexity with respect t o partial orders over subsets of Euclidean spaces. Our results are use d in Hwang, Rothblum and Shepp (1993) to address some optimal assembly problems.