Outer billiards of symplectically self-polar convex bodies

TL;DR

This paper extends the concept of invariant curves in outer billiards from 2D Radon norms to higher dimensions, introducing symplectically self-polar convex bodies as new examples with invariant hypersurfaces.

Contribution

It demonstrates that symplectically self-polar convex bodies in higher dimensions admit invariant hypersurfaces for outer billiards, unlike the classical case where only ellipsoids have caustics.

Findings

Symplectically self-polar convex bodies have invariant hypersurfaces for outer billiards.

Extension of 2D Radon norm properties to higher dimensions.

First non-trivial examples of invariant hypersurfaces in higher-dimensional outer billiards.

Abstract

It is known that $C^1$-smooth strictly convex Radon norms in $\mathbb{R}^2$ can be characterized by the property that the outer billiard map, which corresponds to the unit ball of the norm, has an invariant curve consisting of 4-periodic orbits. In higher dimensions, Radon norms are necessarily Euclidean. However, we show in this paper that the property of existence of an invariant curve of 4-periodic orbits allows a higher-dimensional extension to the class of symplectically self-polar convex bodies. Moreover, this class of convex bodies provides the first non-trivial examples of invariant hypersurfaces for outer billiard map. This is in contrast with conventional Birkhoff billiards in higher dimensions, where it was proved by Berger and Gruber that only ellipsoids have caustics. It is not known, however, if non-trivial invariant hypersurfaces can exist for higher-dimensional Birkhoff billiards.