Directional-Quasi-Convexity, Asymmetric Schur-Convexity and Optimality of Consecutive Partitions

The current paper has the following distinct goals:

1. To generalize standard quasi-convexity to obtain a weaker property of functions that suffices for optimality of extreme points. Specifically, it is required that intersections of the level sets of the function with line segments having direction in a prescribed set are convex.

2. To characterize Schur-convexity for symmetric functions through the above generalization of quasi-convexity, thereby obtaining a previously unknown relationship of Schur-convexity and (standard) convexity. The new characterization is used to extend the definition of Schur-convexity to functions which are not symmetric.

3. To obtain new sufficient conditions for the optimality of consecutive partitions, by using the new definition of Schur-convexity for functions that are not necessarily symmetric.

The conclusion that Schur-convexity is an instance of a convexity property that implies the optimality of extreme points, unifies two approaches that were used in the literature to prove optimality of subsets of the domain of real-valued functions. The two tools—quasi-convexity and Schur-convexity—were previously considered as distinct techniques.