Characterizations of strictly convex sets by the uniqueness of support points. (3rd July 2019)

• Record Type: Journal Article
• Authors: Ha, Truong Xuan Duc; Jahn, Johannes
• Is Part Of: Optimization. Volume 68:Number 7(2019)
• Page Start: 1321
• Page End: 1335
• Publication Date: 2019-07-03
• Subjects: Convex sets -- convex analysis -- support points -- extreme points -- Demyanov difference of sets; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2018.1476513
• Languages: English
• ISSNs: 0233-1934

Abstract

Abstract: This short paper characterizes strictly convex sets by the uniqueness of support points (such points are called unique support points or exposed points) under appropriate assumptions. A class of so-called regular sets, for which every extreme point is a unique support point, is introduced. Closed strictly convex sets and their intersections with some other sets are shown to belong to this class. The obtained characterizations are then applied to set-valued maps and to the separation of a convex set and a strictly convex set. Under suitable assumptions, so-called set-valued maps with path property are characterized by strictly convex images of the considered set-valued map.