DIRECTIONAL-QUASI-CONVEXITY, ASYMMETRIC SCHUR-CONVEXITY AND OPTIMALITY OF CONSECUTIVE PARTITIONS

The current paper has the following distinct goals: 1. To generalize s tandard quasi-convexity to obtain a weaker property of functions that suffices for optimality of extreme points. Specifically, it is require d that intersections of the level sets of the function with line segme nts having direction in a prescribed set are convex. 2. To characteriz e Schur-convexity for symmetric functions through the above generaliza tion of quasi-convexity, thereby obtaining a previously unknown relati onship of Schur-convexity and (standard) convexity. The new characteri zation is used to extend the definition of Schur-convexity to function s 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. T he conclusion that Schur-convexity is an instance of a convexity prope rty that implies the optimality of extreme points, unifies two approac hes that were used in the literature to prove optimality of subsets of the domain of real-valued functions. The two tools-quasi-convexity an d Schur-convexity-were previously considered as distinct techniques.