Continuous Sets and Non-Attaining Fuctionals in Reflexive Banach Spaces

  • Emil Ernst
  • Michel Théra
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 79)


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 phrases

Continuous closed convex set non-attaining functional well-positioned set non-coercive variational inequalities 


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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Emil Ernst
    • 1
  • Michel Théra
    • 2
  1. 1.Laboratory of Modelisation in Mechanics and ThermodynamicsFaculty of Science and Techniques of Saint JeromeSaint JérômeFrance
  2. 2.LacoUniversity of LimogesLimoges CedexFrance

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