Theorem \[t:oszoll\] showing that the action and the volume of a weakly Zoll pair are related through the Zoll polynomial.

-   We formulate the local systolic-diastolic inequality for odd-symplectic forms in Section \[sec:conjsys\]. We prove it for quasi-autonomous H-forms (Proposition \[p:qa1\] corresponding to Proposition \[p:qaos\]) or when the real Euler class of the bundle vanishes (end of Subsection \[ss:triv\] corresponding to Theorem \[t:triv\] and Proposition \[p:hamsys\]).

**Acknowledgements.** This work is part of a project in the Collaborative Research Center *TRR 191 - Symplectic Structures in Geometry, Algebra and Dynamics* funded by the DFG. It was initiated when the authors worked together at the University of Münster and partially carried out while J.K. was affiliated with the Ruhr-University Bochum. We thank Peter Albers, Kai Zehmisch, and the University of Münster for having provided an inspiring academic environment. We are also grateful to Michael Kapovich for the characterisation contained in Lemma \[l:3state1\]. G.B. would like to express his gratitude to Hans-Bert Rademacher and the whole Differential Geometry group at the University of Leipzig. J.K. was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1801-01.

Notations {#notations .unnumbered}
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In the following discussion $(\Sigma,\mathfrak o_\Sigma)$ is a connected oriented closed manifold 