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Reversed isoperimetric inequality for β in outer billiards

Prove a reversed isoperimetric-type inequality for the Mather β-function of outer billiards under the existence of a rotational invariant curve, extending known cases for rotation numbers 1/3 and 1/4.

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Background

The authors established an isoperimetric-type inequality for β in several models, with outer billiards showing exceptional behavior (sign reversal) in certain symmetric cases.

They ask to extend reversed-inequality results to general outer billiards assuming the presence of a rotational invariant curve.

References

In this section we formulate natural open questions related to the results discussed in previous sections. (1) Can one prove the reversed isoperimetric inequality for the Mather $\beta$-function of outer billiards, assuming the existence of a rotational invariant curve. As we mentioned above, this is the case for rotation numbers $\frac 13, \frac 14$.

— Integrable Billiards and Related Topics (2510.03790 - Bialy et al., 4 Oct 2025) in Section 9 (Open questions), Subsection Mather β-function, item (1)