Directional quasi-convexity, asymmetric Schur-convexity and optimality of consecutive partitions. (English) Zbl 0866.26008
Summary: 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.
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.
MSC:
| 26B25 | Convexity of real functions of several variables, generalizations |
| 52A41 | Convex functions and convex programs in convex geometry |
| 26D15 | Inequalities for sums, series and integrals |
