  The aim of this paper is to formulate a local systolic inequality for
odd-symplectic forms (also known as Hamiltonian structures) and to establish it
in some basic cases. Let $\Omega$ be an odd-symplectic form on an oriented
closed manifold $\Sigma$ of odd dimension. We say that $\Omega$ is Zoll if the
trajectories of the flow given by $\Omega$ are the orbits of a free
$S^1$-action. After defining the volume of $\Omega$ and the action of its
periodic orbits, we prove that the volume and the action satisfy a polynomial
equation, provided $\Omega$ is Zoll. This builds the equality case of a
conjectural systolic inequality for odd-symplectic forms close to a Zoll one.
We prove the conjecture when the $S^1$-action yields a flat $S^1$-bundle or
$\Omega$ is quasi-autonomous. In particular the conjecture is established in
dimension three.
  This new inequality recovers the contact systolic inequality as well as the
inequality between the minimal action and the Calabi invariant for Hamiltonian
isotopies $C^1$-close to the identity on a closed symplectic manifold.
Applications to the study of periodic magnetic geodesics on closed orientable
surfaces is given in the companion paper available at arXiv:1902.01262.
