A dual mapping associated to a closed convex set and some subdifferential properties
DOI:
https://doi.org/10.24193/subbmath.2024.1.06Keywords:
Distance function associated to a set, $\varepsilon $- subdifferential, best approximation element, $\varepsilon $- monotonicity, separating hyperplaneAbstract
In this paper we establish some properties of the mapping \(\left( x,d\right) \mapsto D_{C}\left( x;d\right)\) that associates to every element \(x\) of a linear normed space \(X\) the set of linear continuous functionals of norm \(d\geq 0\) and which separates the closed ball \(B\left( x;d\right)\) from a closed convex set \(C\subset X\).
Using this mapping we give links with other important concepts in convex analysis (\(\varepsilon\)-approximation element, \(\varepsilon\)-subdifferential of distance function, duality mapping, polar cone). Thus, we establish a dual characterization of \(\varepsilon\)-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 \(D_{C}\).
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