Billiards and Quantitative Symplectic Geometry — Misha Bialy — Problem 2

Definitions and mathematical statement

Let $N:{\mathbb{R}}^2\to[0,\infty)$ be a possibly asymmetric Minkowski norm: $N(v)=0$ if and only if $v=0$, $N(tv)=tN(v)$ for $t\geq0$, and $N(u+v)\leq N(u)+N(v)$. The identity $N(-v)=N(v)$ is not assumed. Set \[ B_N=\{v\in{\mathbb{R}}^2:N(v)\leq1\}. \] Consider the billiard table whose boundary is $\partial B_N$, in the setting of the source's smooth convex boundary construction. Parametrize this boundary by a periodic map $\gamma$ with $N(\dot\gamma(s))=1$. The Minkowski billiard associated with $N$ has generating function \[ L(s_0,s_1)=N\left(\gamma(s_1)-\gamma(s_0)\right). \] Its trajectories satisfy the variational principle associated with \[ \mathcal L\{s_n\}=\sum_n L(s_n,s_{n+1}). \] Here stationarity can be expressed using finite segments with fixed endpoints; the displayed sum need not be interpreted as a convergent infinite series. The source describes the resulting billiard as a twist map. For a Euclidean norm, this construction specializes to the Birkhoff billiard.

Question (communicated by Yaron Ostrover). For every possibly asymmetric Minkowski norm $N$ for which this billiard construction applies, does total integrability of the Minkowski billiard in $B_N$, using the same norm $N$ in the generating function, imply that $N$ is Euclidean? With the usual meaning of a Euclidean norm, the proposed conclusion is \[ \exists\,Q\in{\mathbb{R}}^{2\times2}\quad Q^{\mathsf T}=Q,\qquad v^{\mathsf T}Qv>0\ \text{for every }v\ne0,\qquad N(v)=\sqrt{v^{\mathsf T}Qv}\ \text{for every }v\in{\mathbb{R}}^2. \] Unresolved qualifications. The supplied record does not define total integrability or specify its required regularity and phase-space coverage. It also does not give the precise smoothness and strict-convexity hypotheses on $N$ and $\partial B_N$ needed for the asserted twist-map construction. These conditions must be recovered from the source before the implication has a fully specified domain. The inner-product interpretation of "Euclidean" above should likewise be checked against the source's conventions.

The supplied record states that the question was answered affirmatively in reference [IMRN] for billiard tables with a finite symmetry group. It does not specify the symmetry group or the hypotheses of that result. This restricted affirmative assertion is retained as supplied literature context, without independent verification; the registry labels the problem open.

Short English statement

If the Minkowski billiard inside the unit ball of its own possibly asymmetric norm is totally integrable, must that norm come from an inner product? The source reports an affirmative answer for tables with a finite symmetry group. The exact integrability, regularity, and symmetry hypotheses remain to be checked.

Sources

• Local UnsolvedMath registry, permanent ID $800467$, source problem code catalogue entry; title: "Billiards and Quantitative Symplectic Geometry --- Misha Bialy --- Problem 2." Supplied status: open; this is a registry label, not an independently verified literature assessment. Upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Albach, Barraud et al., Open Problems in Billiards and Quantitative Symplectic Geometry (2026), Misha Bialy --- Problem 2. Supplied source URL: https://arxiv.org/abs/2602.12896. The record reports access on 2026-10-16 and extraction from arXiv LaTeX; these are supplied provenance details, not a claim of independent access or verification. Contributor: Misha Bialy. The question is credited as communicated by Yaron Ostrover.
• Reference [GUTKIN2002277], attributed in the supplied statement to Gutkin and Tabachnikov, for the introduction and study of Minkowski and Finsler billiards. No full bibliographic entry or reference-specific URL is supplied.
• Reference [Artstein_Avidan_2014], attributed in the supplied statement to Artstein-Avidan, Karasev, and Ostrover, for work connecting this model with Mahler's conjecture. No full bibliographic entry or reference-specific URL is supplied.
• Reference [IMRN], cited by the supplied statement for an affirmative answer for billiard tables with a finite symmetry group. Authors, title, date, precise hypotheses, and a reference-specific URL are not supplied.