  Given a strictly concave rational PL function $\phi$ on a complete
$n$-dimensional fan $\Sigma$, we construct an exact symplectic structure of
finite volume on $(\mathbb{C}^\times)^n$ and a family of functions
$H_{\phi,\epsilon}$ called polyhedral Hamiltonians. We prove that for each
$\epsilon$ the one-periodic orbits of $H_{\phi,\epsilon}$ come in families
corresponding to finitely many primitive lattice points of $\Sigma$ and
determine their topology. When $\phi$ is negative on the rays of $\Sigma$, we
show that the level sets of polyhedral Hamiltonians are hypersurfaces of
contact type. As a byproduct, this construction provides a dynamical model for
the singularities of toric varieties obtained as degenerations of Fano
manifolds in any dimension via Okounkov bodies.
