A Variational Principle in Reflexive Spaces with Kadec-Klee Norm

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