Continuous Sets and Non-Attaining Fuctionals in Reflexive Banach Spaces
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.
Key words and phrasesContinuous closed convex set non-attaining functional well-positioned set non-coercive variational inequalities
Unable to display preview. Download preview PDF.
- G. Del Piero, A Condition for Statical Admissibility in Unilateral Structural Analysis, Theoretical and Numerical Non-smooth Mechanics, International Colloquium in honor of the 80th birthday of Jean Jacques Moreau, 17–19 November 2003, Montpellier, FranceGoogle Scholar
- R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series 28, Princeton University Press, 1970.Google Scholar
- E. Zeidler, Nonlinear Functional Analysis and its Applications II, Springer-Verlag, 1990.Google Scholar