On the structure of minimizers of causal variational principles in the non-compact and equivariant settings

We derive the Euler-Lagrange equations for minimizers of causal variational principles in the non-compact setting with constraints, possibly prescribing symmetries. Considering first variations, we show that the minimizing measure is supported on the intersection of a hyperplane with a level set of a function which is homogeneous of degree two. Moreover, we perform second variations to obtain that the compact operator representing the quadratic part of the action is positive semi-definite. The key ingredient for the proof is a subtle adaptation of the Lagrange multiplier method to variational principles on convex sets.