Group invariant variational principles

In this paper we introduce a group invariant version of the well-known Ekeland variational principle. To achieve this, we define the concept of convexity with respect to a group and establish a version of the theorem within this framework. Additionally, we present several consequences of the group invariant Ekeland variational principle, including Palais-Smale minimizing sequences, the Brønsted-Rockafellar theorem, and a characterization of the linear and continuous group invariant functionals space. Moreover, we provide an alternative proof of the Bishop-Phelps theorem and proofs for the group-invariant Hahn-Banach separating theorems. Finally, we discuss some implications and applications of these results.