On Hamiltonian projective billiards on boundaries of products of convex bodies

TL;DR

This paper investigates Hamiltonian projective billiards on boundaries of convex bodies' products, revealing that the projective reflection law corresponds to ellipsoids, and explores implications for symplectic and convex geometry conjectures.

Contribution

It characterizes when the projective billiard reflection law applies, showing it occurs only for ellipsoids, and connects this to affine equivalence with Euclidean billiards and Finsler billiards.

Findings

Projective billiard reflection law holds iff T is an ellipsoid.

All T-billiards are affine equivalent to Euclidean billiards under this law.

Results relate to symplectic isoperimetric and Mahler conjectures.

Abstract

Let $K\subset\mathbb R^n_q$, $T\subset\mathbb R^n_p$ be two bounded strictly convex bodies (open subsets) with $C^6$-smooth boundaries. We consider the product $\overline K\times\overline T\subset\mathbb R^{2n}_{q,p}$ equipped with the standard symplectic form $\omega=\sum_{j=1}^ndq_j\wedge dp_j$. The $(K,T)$-billiard orbits are continuous curves in the boundary $\partial(K\times T)$ whose intersections with the open dense subset $(K\times\partial T)\cup(\partial K\times T)$ are tangent to the characteristic line field given by kernels of the restrictions of the symplectic form $\omega$ to the tangent spaces to the boundary. For every $(q,p)\in K\times \partial T$ the characteristic line in $T_{(q,p)}\mathbb R^{2n}$ is directed by the vector $(\vec n(p),0)$, where $\vec n(p)$ is the exterior normal to $T_p\partial T$, and similar statement holds for $(q,p)\in\partial K\times T$. The projection of each $(K,T)$-billiard orbit to $K$ is an orbit of the so-called $T$-billiard in $K$. In the case, when $T$ is centrally-symmetric, this is the billiard in $\mathbb R^n_q$ equipped with Minkowski Finsler structure "dual to $T$", with Finsler reflection law introduced in a joint paper by S.Tabachnikov and E.Gutkin in 2002. Studying $(K,T)$-billiard orbits is closely related to C.Viterbo's Symplectic Isoperimetric Conjecture (recently disproved by P.Haim-Kislev and Y.Ostrover) and the famous Mahler Conjecture in convex geometry. We study the special case, when the $T$-billiard reflection law is the projective law introduced by S.Tabachnikov, i.e., given by projective involutions of the projectivized tangent spaces $T_q\mathbb R^n$, $q\in\partial K$. We show that this happens, if and only if $T$ is an ellipsoid, or equivalently, if all the $T$-billiards are simultaneously affine equivalent to Euclidean billiards. As an application, we deduce analogous results for Finsler billiards.