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 convex caustic is a convex curve inside the table such that any billiard segment tangent to it remains tangent after reflection. Symmetry about $l$ means invariance under reflection in that line; smoothness is required of the table, while the question does not add smoothness of the caustic.

Statement recorded in the supplied source.

Let $\gamma$ be a smooth convex plane billiard table symmetric about an axis $l$, and let $C$ be a convex caustic. Must $C$ be symmetric about $l$? Prove this or give a counterexample. [S2]

Short English statement

Must every convex caustic inherit a reflection symmetry of its smooth convex billiard table? Prove this or find a counterexample.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5200005 (AMR-051-0005). 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, Misha Bialy, Problem 5, p. 4. https://arxiv.org/pdf/2110.10750v2.