A dual mapping associated to a closed convex set and some subdifferential properties

Abstract

In this paper, we establish some properties of the multivalued mapping (x, d) ⇒ DC (x; d) that associates to every element x of a linear normed space X the set of linear continuous functionals of norm d ≥ 0 and which separates the closed ball B (x; d) from a closed convex set C ⊂ X. Using this mapping we give links with other important concepts in convex analysis (ε-approximation element, ε-subdifferential of distance function, duality mapping, polar cone). Thus, we establish a dual characterization of ε-approximation elements with respect to a nonvoid closed convex set as a generalization of a known result of Garkavi. Also, we give some properties of univocity and monotonicity of mapping DC. Mathematics Subject Classification (2010): 32A70, 41A65, 46B20, 46N10. Received 18 May 2023; Accepted 27 November 2023