Journal of Convex Analysis 16 (2009), No. 1, 211--226
Copyright Heldermann Verlag 2009
A Variational Principle in Reflexive Spaces with Kadec-Klee Norm
Marián Fabian
Mathematical Institute, Czech Academy of Sciences, Zitná 25, 115 67 Praha 1, Czech Republic
fabian@math.cas.cz
Julian Revalski
Inst. of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Street Block 8, 1113 Sofia, Bulgaria
and: Laboratoire AOC, Dép. de Mathématiques et Informatique, Université des Antilles et de la Guyane, BP 592 - Campus de Fouillole, 97157 Pointe-à-Pitre Guadeloupe, France
julian.revalski@univ-ag.fr
We prove a variational principle in reflexive Banach spaces X with Kadec-Klee norm, which asserts that any Lipschitz (or any proper lower semicontinuous bounded from below extended real-valued) function in X can be perturbed with a parabola in such a way that the perturbed function attains its infimum (even more can be said -- the infimum is well-posed). In addition, we have genericity of the points determining the parabolas. We prove also that the validity of such a principle actually characterizes the reflexive spaces with Kadec-Klee norm. This principle turns out to be an analytic counterpart of a result of K.-S. Lau on nearest points.
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