On Hamiltonian projective billiards on boundaries of products of convex bodies

Let K⊂ Rⁿq, T⊂ Rⁿₚ be two bounded strictly convex bodies (open subsets) with C⁶-smooth boundaries. We consider the product K× T⊂ R²ⁿp,q equipped with the standard symplectic form ω=∑ⱼ₌₁ⁿdqⱼ dpⱼ. The $(K,T)$-billiard orbits are continuous curves in the boundary ∂(K× T) whose intersections with the open dense subset (K×∂ T)∪(∂ K× T) are tangent to the characteristic line field given by kernels of the restrictions of the symplectic form ω to the tangent spaces to the boundary. For every (q,p)∈ K× ∂ T the characteristic line in T(q,p) R²ⁿ is directed by the vector ( n(p),0), where n(p) is the exterior normal to Tₚ∂ T, and similar statement holds for (q,p)∈∂ K× 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 Rⁿq 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 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 Tq Rⁿ, q∈∂ 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.