  The periodic Toda lattice with $N$ sites is globally symplectomorphic to a
two parameter family of $N-1$ coupled harmonic oscillators. The action
variables fill out the whole positive quadrant of $\R^{N-1}$. We prove that in
the interior of the positive quadrant as well as in a neighborhood of the
origin, the Toda Hamiltonian is strictly convex and therefore Nekhoroshev's
theorem applies on (almost) all parts of phase space.
