Outer billiards of symplectically self-polar convex bodies

TL;DR

This work extends the planar Radon-norm characterization to higher dimensions by introducing symplectically self-polar convex bodies $X\subset\mathbb{R}^{2n}$ with symplectic polar $X^\omega = JX^\circ$. It defines a boundary map $f$ on $\partial X$ via $\omega(x,f(x))=1$ (with $f^2=-\mathrm{id}$) and proves the outer billiard map possesses an invariant hypersurface consisting of centrally symmetric 4-periodic orbits for $C^1$-smooth $X$, while in the $C^2$-smooth case a converse holds (up to scaling). The paper further shows that the set $Y=\{x+f(x): x\in\partial X\}$ is $T$-invariant and forms a topological sphere bounding a star-shaped region, providing the first non-trivial invariant hypersurfaces in outer billiards beyond planar caustics. It also constructs concrete examples of symplectically self-polar bodies with $C^1$ and $C^\infty$ boundaries, including non-Euclidean ones, via $l_2$-sums and smoothing techniques. These results illuminate the geometry of outer billiards in higher dimensions and raise questions about uniqueness, area constructions, and the existence of unbounded orbits.

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.

Figures (2)

• Figure 1: The symplectic outer billiard map in the plane
• Figure 2: Area construction applied to $Y$ with the area parameter $\frac{A-4}{4}$ .

Theorems & Definitions (24)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof