  We study some particular cases of Viterbo's conjecture relating volumes of
convex bodies and actions of closed characteristics on their boundaries,
focusing on the case of a Hamiltonian of classical mechanical type, splitting
into summands depending on the coordinates and the momentum separately. We
manage to establish the conjecture for sublevel sets of convex $2$-homogeneous
Hamiltonians of this kind in several particular cases. We also discuss open
cases of this conjecture.
