In Newtonian gravity galaxies are typically modeled as steady states of the Vlasov-Poisson system. It is well-known that spherically symmetric steady states of the Vlasov-Poisson system can be obtained as minimizers of an energy-Casimir functional. This has played a crucial role for the stability theory that has been developed in that case. It is well-known that there is no analogue stability theory in the relativistic case, i.e. for the Einstein-Vlasov system, mainly due to lack of compactness. In this talk I will briefly discuss the stability results in the Newtonian case and then I will present a recent result, together with Markus Kunze, where we show compactness of minimizing sequences to a particle number-Casimir functional which implies the existence of a minimizer to this functional. As a consequence of our proof, a condition arises which we conjecture is sufficient for nonlinear stability