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.

The table is bounded and strictly convex with smooth boundary. A closed caustic is an interior closed curve for which reflection preserves tangency of billiard chords. The two caustics are nested. In addition to being caustics for the original table, the inner one must be a caustic for the billiard whose boundary is the outer caustic. Only these two caustics are assumed.

Statement recorded in the supplied source.

Suppose a bounded strictly convex planar billiard with smooth boundary has two nested closed caustics such that the smaller is itself a caustic for the billiard inside the larger. Must its boundary be an ellipse? [S2]

Short English statement

Does the existence of two nested caustics, with the inner also a caustic inside the outer, force the original smooth convex table to be an ellipse?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5200007 (AMR-051-0007). 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, Alexey Glutsyuk, Problem 1 (question of Konstantin Khanin, August 2019), pp. 5--6. https://arxiv.org/pdf/2110.10750v2.