  Let M be a closed symplectic manifold, and let | | be a norm on the space of
all smooth functions on M, which are zero-mean normalized with respect to the
canonical volume form. We show that if | | is dominated from above by the
L-Infinity-norm, and | | is invariant under the action of Hamiltonian
diffeomorphisms, then it is also invariant under all volume preserving
diffeomorphisms. We also prove that if | | is, additionally, not equivalent to
the L-Infinity-norm, then the induced Finsler metric on the group of
Hamiltonian diffeomorphisms on M vanishes identically.
