Journal of Convex Analysis 16 (2009), No. 1, 211--226
Copyright Heldermann Verlag 2009
A Variational Principle in Reflexive Spaces with Kadec-Klee Norm
Mathematical Institute, Czech Academy of Sciences, Zitná 25, 115 67 Praha 1, Czech Republic
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
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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