On polynomially integrable planar outer billiards and curves with symmetry property

TL;DR

This paper proves that any polynomially integrable planar outer convex billiard must be an elliptic curve, establishing a significant geometric classification result.

Contribution

It demonstrates that polynomial integrability in planar outer billiards implies the billiard boundary is necessarily an ellipse, a novel classification insight.

Findings

Polynomial integrability implies elliptic boundary curves.

Outer billiards with polynomial integrability are exclusively elliptic.

The result narrows the class of integrable outer billiards to ellipses.

Abstract

We show that every polynomially integrable planar outer convex billiard is elliptic.