Open Problems on Billiards and Geometric Optics

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A ray is an oriented Euclidean line before entering a mirror system. Trapping means that after entry its successive optical reflections keep it in the trapping domain and prevent escape. The line space $\mathcal L$ has dimension $2n-2$ and its standard symplectic form. A local normal family consists of rays perpendicular to a hypersurface; it is a Lagrangian submanifold of line space of dimension $n-1$. A non-normal family lacks this property.

Statement recorded in the supplied source.

In $\mathbb{R}^n$, the space $\mathcal{L}$ of oriented lines has dimension $2n-2$ and a natural symplectic structure. Normal families of rays form Lagrangian submanifolds and can be trapped by mirrors. What is the greatest dimension of a family of rays that can be trapped? In $\mathbb{R}^3$, can a non-normal two-parameter family of rays be trapped? [S2]

Short English statement

Find the maximum dimension of a family of rays that mirrors can trap in Euclidean space; in three dimensions, decide whether a non-normal two-parameter family can be trapped.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5200013 (AMR-051-0013). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 M. Bialy, C. Fierobe, A. Glutsyuk, M. Levi, A. Plakhov and S. Tabachnikov, Open problems on billiards and geometric optics (2021), arXiv:2110.10750v2, Serge Tabachnikov, Problem 1, pp. 9--10. https://arxiv.org/pdf/2110.10750v2.