On Hamiltonian Projective Billiards on Boundaries of Products of Convex Bodies

Let $K\subset \mathbb R^n_q$ and $T\subset \mathbb R^n_p$ 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 the 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 outward normal to $T_p\partial T$, and a 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 the 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 Tabachnikov, i.e., is 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.