On Hamiltonian projective billiards on boundaries of products of convex bodies

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.