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 $$\,\overline{\negthinspace K}\times\overline T\subset\mathbb R^{2n}_{q,p}$$ 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.