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Equality of EHZ and cylindrical capacities on convex domains

Determine whether the Ekeland–Hofer–Zehnder capacity coincides with the cylindrical capacity on the class of convex domains in R^{2n}.

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Background

The authors show that the Gromov width and the Ekeland–Hofer–Zehnder capacity can differ on convex domains, leaving multiple a priori distinct capacities on this class. In dimension four for dynamically convex domains, results imply that the first embedded contact homology capacity equals the cylindrical capacity, but this does not resolve the relationship between the EHZ and cylindrical capacities on convex domains.

For specific examples (e.g., the pentagon product K × T considered in the paper), the EHZ and cylindrical capacities coincide, but the general equality question remains open.

References

While Theorem~\ref{counterexample_thm} demonstrates that the Gromov width differs from the EHZ capacity, it remains an open question whether the EHZ capacity coincides with the cylindrical capacity for convex domains (note that in the case of the products of pentagons $K \times T$ mentioned above, a simple computation shows that these two capacities indeed coincide).

— A Counterexample to Viterbo's Conjecture (2405.16513 - Haim-Kislev et al., 26 May 2024) in Discussion and Open Questions (v)