Abstract
In this paper we prove, in the framework of reflexive Banach spaces, that a linear and continuous functional f achieves its supremum on every small ε -uniform perturbation of a closed convex set C containing no lines, if and only if f belongs to the norm-interior of the barrier cone of C. This result is applied to prove that every closed convex subset C of a reflexive Banach space X which contains no lines is continuous if and only if every small ε -uniform perturbation of C does not allow non-attaining linear and continuous functionals. Finally, we define a new class of non-coercive variational inequalities and state a corresponding open problem.
The research of Michel Théra has been supported by NATO Collaborative Linkage Grant 978488.
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Ernst, E., Théra, M. (2005). Continuous Sets and Non-Attaining Fuctionals in Reflexive Banach Spaces. In: Giannessi, F., Maugeri, A. (eds) Variational Analysis and Applications. Nonconvex Optimization and Its Applications, vol 79. Springer, Boston, MA. https://doi.org/10.1007/0-387-24276-7_22
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DOI: https://doi.org/10.1007/0-387-24276-7_22
Publisher Name: Springer, Boston, MA
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