  We consider the three dimensional gravitational Vlasov-Poisson (GVP) system
in both classical and relativistic cases. The classical problem is subcritical
in the natural energy space and the stability of a large class of ground states
has been derived by various authors. The relativistic problem is critical and
displays finite time blow up solutions. Using standard concentration
compactness techniques, we however show that the breaking of the scaling
symmetry allows the existence of stable relativistic ground states. A new
feature in our analysis which applies both to the classical and relativistic
problem is that the orbital stability of the ground states does not rely as
usual on an argument of uniqueness of suitable minimizers --which is mostly
unknown-- but on strong rigidity properties of the transport flow, and this
extends the class of minimizers for which orbital stability is now proved.
