Symplectic billiards as Minkowski billiards

We establish a connection between Minkowski billiards and symplectic billiards, two classes of dynamical systems that have been studied largely independently. We show that the Minkowski billiard map can be described in symplectic terms via reduction from the canonical symplectic structure on $V \times V^*$, and that symplectic billiards can be viewed as a ``square root'' of a symplectic version of Minkowski billiards. As an application, we recover several known results on symplectic billiards from the more general Minkowski setting, and extend some of them to higher dimensions and to periodic orbits of even period. In particular, we prove the existence of at least $(r-1)(n-1)$ $2r$-periodic symplectic billiard orbits in dimension $2n$.