On Hamiltonian projective billiards on boundaries of products of convex bodies

Abstract

Let $K\subset {\mathbb{R}}_q^n$, $T\subset {\mathbb{R}}_p^n$ be two bounded strictly convex bodies (open subsets) with $C^6$-smooth boundaries. We consider the product $\overline{K}\times \overline{T}\subset {\mathbb{R}}_{q,p}^{2n}$ equipped with the standard symplectic form $\omega ={\sum }_{j=1}^{n}d{q}_j\wedge d{p}_j$. 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 ${\mathbb{R}}_q^n$ 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.

26 pages, 4 figures. Minor corrections. To appear in Proc. Steklov Inst. Math

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• If a Minkowski billiard is projective, it is the standard billiard