A local systolic-diastolic inequality in contact and symplectic geometry

Let $\mathrm{\Sigma }$ be a connected closed three-manifold, and let $t_{\mathrm{\Sigma }}$ be the order of the torsion subgroup of $H_1(\mathrm{\Sigma };\mathbb{Z})$. For a contact form $\alpha$ on $\mathrm{\Sigma }$, we denote by $\mathrm{V}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{e}(\alpha )$ the contact volume of $\alpha$, and by $T_{\min }(\alpha )$ and $T_{\max }(\alpha )$ the minimal period and the maximal period of prime periodic orbits of the Reeb flow of $\alpha$ respectively. We say that $\alpha$ is Zoll if its Reeb flow generates a free $S^1$-action on $\mathrm{\Sigma }$. We prove that every Zoll contact form ${\alpha }_{\ast }$ on $\mathrm{\Sigma }$ admits a $C^3$-neighbourhood $\mathcal{U}$ in the space of contact forms such that $$t_{\mathrm{\Sigma }}{T}_{\min }(\alpha {)}^2\le \mathrm{V}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{e}(\alpha )\le t_{\mathrm{\Sigma }}{T}_{\max }(\alpha {)}^2,\mathrm{\forall }\alpha \in \mathcal{U},$$ and any of the equalities holds if and only if $\alpha$ is Zoll. We extend the above picture to odd-symplectic forms $\mathrm{\Omega }$ on $\mathrm{\Sigma }$ of arbitrary odd dimension. We define the volume of $\mathrm{\Omega }$, which generalises both the contact volume and the Calabi invariant of Hamiltonian functions, and the action of closed characteristics of $\mathrm{\Omega }$, which generalises both the period of periodic Reeb orbits and the action of fixed points of Hamiltonian diffeomorphisms. We say that $\mathrm{\Omega }$ is Zoll if its characteristics are the orbits of a free $S^1$-action on $\mathrm{\Sigma }$. We prove that the volume and the action of a Zoll odd-symplectic form satisfy a certain polynomial equation. This builds the equality case of a conjectural local systolic-diastolic inequality for odd-symplectic forms, which we establish in some cases. This inequality recovers the inequality between the minimal action and the Calabi invariant of Hamiltonian isotopies $C^1$-close to the identity on a closed symplectic manifold, as well as the local contact systolic-diastolic inequality above. Finally, applications to magnetic geodesics are discussed.