# Notes on the Ekeland Variational Principle

Beer, Gerald

### Abstract

We show that the weak Ekeland variational principle is itself derivative of a geometric principle for Lipschitz real-valued functions. We use this principle to directly obtain an omnibus result regarding the nonempty intersection of a decreasing sequence of nonempty closed sets. Both the weak Ekeland principle and our principle for Lipschitz functions are characteristic of completeness of the underlying metric. We also characterize a class of metric spaces lying strictly between the compact and complete metric spaces by an Ekeland-type principle: the class of metric spaces *TeX Embedding failed!* on which each continuous functions with values in an arbitrary metric space is uniformly continuous.

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