Majorization and Schur convexity with respect to partial orders

Majorization and Schur convexity constitute an important tool for establishing inequalities, in particular, due to the Schur-Ostrowski Theorem which provides a simple characterization of Schur convex functions through local two-coordinate conditions. However, the usefulness of the approach is limited by two stringent requirements. First, every Schur convex function must be symmetric. Second, the necessary and sufficient conditions for Schur convexity involve every pair of coordinates on its (symmetric) domain. Several attempts have been made to relax these two requirements. In particular, a more general concept of ‘majorization with respect to partial orders’ was introduced to prepare for such tasks, but it did not capture the classic theory for symmetric functions. In the current paper the authors obtain the desired generalization by developing a theory of majorization and Schur convexity with respect to partial orders over subsets of Euclidean spaces. The present results are used in Hwang, Rothblum and Shepp to address some optimal assembly problems.