The talk aims to review some recent results on the long-time
stability of elliptic equilibria of Hamiltonian systems. These results have
been obtained within the so called Nekhoroshev theory under the assumption
that the low order Birkhoff normal forms have certain properties. A key
notion is that of "directional quasi-convexity" (DQC) of the fourth order
Birkhoff normal form, which has revealed to be very powerful in the study of
several classical applications (e.g., equilateral Lagrangian points of the
restricted spatial three body problem; Riemann ellipsoids). After reviewing
the theory, we shall focus on the issue of the numerical detection of DQC in
systems which, like e.g. the top toy known as ``levitron'', have five or more
degreees of freedom.