Rigidity of a totally integrable Minkowski billiard in its own unit ball
Let $N : \mathbb{R}^2 \to [0, \infty)$ be a norm, smooth away from $0$, such that $N^2$ has positive-definite Hessian there. Put $K = \{v : N(v) \le 1\}$ and parametrize $\partial K$ by $N$-arclength $\gamma(s)$. The Minkowski billiard in $K$ for the norm $N$ is the twist map on ordered pairs of distinct boundary points generated by $L(s,t) = N(\gamma(t) - \gamma(s))$: successive parameters satisfy $$\partial_2 L(s_{j-1}, s_j) + \partial_1 L(s_j, s_{j+1}) = 0.$$ For a Euclidean norm, $K$ is a disk and the billiard is the circular one, whose phase annulus is foliated by invariant curves. Bialy proved that a convex Birkhoff billiard (the Euclidean norm, on any table) with this property is a circle. Bialy and Tsodikovich proved that if $\partial K$ is $C^2$ and strictly convex, $K$ is invariant under a linear map of order at least $3$, and the Minkowski billiard in $K$ for the norm of $K$ is totally integrable on the whole phase cylinder, then $\partial K$ is an ellipse.
Problem. If the phase annulus of this billiard map is foliated by continuous invariant simple curves, each winding once around the annulus, must $N(v) = \sqrt{v^{T} Q v}$ for some positive-definite symmetric matrix $Q$?
Source: Ostrover; communicated in Bialy and Mironov (2025), §9.5, Question 1, and in Bialy's contribution to Albach et al. (2026), for Minkowski norms that need not be symmetric. Stated for symmetric norms by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DS — Dynamical Systems
- Secondary
- MG
- Keywords
- AIM Open Applied Problems, integrable billiards, Minkowski billiards, twist maps
References
@misc{bialy2025,
author = {Bialy, Misha and Mironov, Andrey E.},
title = {Integrable billiards and related topics},
year = {2025},
eprint = {2510.03790},
archivePrefix = {arXiv},
note = {Submitted to the proceedings of ICM 2026},
}
@misc{albach2026,
author = {Albach, Bernhard and Barraud, Jean-Fran{\c{c}}ois and Bialy, Misha and Bimmermann, Johanna and Ch{\'a}vez C{\'a}liz, Ana and Damian, Mihai and Deschamps, Lina and Hryniewicz, Umberto and Humili{\`e}re, Vincent and Khesin, Boris and Maier, Levin and Moreno, Agustin and Oancea, Alexandru and Paris-Romaskevich, Olga and Sorrentino, Alfonso and Tabachnikov, Serge},
title = {Open problems in billiards and quantitative symplectic geometry},
year = {2026},
eprint = {2602.12896},
archivePrefix = {arXiv},
}
@article{bialy2023,
author = {Bialy, Misha and Tsodikovich, Daniel},
title = {Billiard tables with rotational symmetry},
journal = {Int. Math. Res. Not. IMRN},
volume = {2023},
number = {5},
year = {2023},
pages = {3970--4003},
doi = {10.1093/imrn/rnab366},
eprint = {2106.06956},
archivePrefix = {arXiv},
}
@article{bialy1993,
author = {Bialy, Misha},
title = {Convex billiards and a theorem by {E}. {H}opf},
journal = {Math. Z.},
volume = {214},
number = {1},
year = {1993},
pages = {147--154},
doi = {10.1007/BF02572397},
}
@misc{aim-self-ball-minkowski-integrability,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Rigidity of a totally integrable Minkowski billiard in its own unit ball}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/252-self-ball-minkowski-integrability.md},
note = {Entry 252 at commit a153f46, 2026-10-13},
}
Resolution claims
Let a smooth strongly convex norm on the plane define both a billiard table and the length of its chords. We prove that if the entire billiard phase annulus is foliated by continuous essential invariant curves, then the norm comes from an inner product. No finite symmetry beyond central symmetry is assumed. The proof retains the support point of each …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Open → Solution claimed | The registry's curators | Total integrability of a Minkowski billiard in its unit ball v1 recorded from AIM @ 24edaad, paper/minkowski_integrability.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00552, "Rigidity of a totally integrable Minkowski billiard in its own unit ball", clopen.solutions.