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Differentiability threshold for the Mather β-function in centrally-symmetric outer billiards

Ascertain whether, for a centrally symmetric outer billiard table, differentiability of the Mather β-function on (0,1/4] or on [1/4,1/2) suffices (in place of differentiability on (0,1/2)) to force the boundary to be an ellipse.

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Background

From total integrability results, differentiability of the β-function on (0,1/2) suffices to imply that the outer billiard boundary is an ellipse.

The authors ask if the required differentiability interval can be reduced in the centrally symmetric case, mirroring sharper results known for Birkhoff billiards.

References

It is an open question whether for a centrally-symmetric outer billiard table one can relax differentiability of the Mather $\beta$-function on $(0,1/2)$, to require the differentiability on $(0,1/4]$ or on $[1/4, 1/2)$.

— Integrable Billiards and Related Topics (2510.03790 - Bialy et al., 4 Oct 2025) in Section 6 (Mather β-function and integrability), after the outer billiard rigidity via β-function