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.