Outer Billiard - Open Questions

Open Questions

Outer billiards is a subject still in its beginning phase. Most problems are still unsolved. Here are some open problems in the area.

• Show that outer billiards relative to almost every convex polygon has unbounded orbits.

• Show that outer billiards relative to a regular polygon has almost every orbit periodic. The cases of the equilaterial triangle and the square are trivial, and Tabachnikov answered this for the regular pentagon. These are the only cases known.

• more broadly, characterize the structure of the set of periodic orbits relative to the typical convex polygon.

• understand the structure of periodic orbits relative to simple shapes in the hyperbolic plane, such as small equilateral triangles.