Billiard tables with rotational symmetry

We generalize the following simple geometric fact: the only centrally symmetric convex curve of constant width is a circle. Billiard interpretation of the condition of constant width reads: a planar curve has constant width, if and only if, the Birkhoff billiard map inside the planar curve has a rotational invariant curve of $2$-periodic orbits. We generalize this statement to curves that are invariant under a rotation by angle $\frac{2\pi}{k}$, for which the billiard map has a rotational invariant curve of $k$-periodic orbits. Similar result holds true also for Outer billiards and Symplectic billiards. Finally, we consider Minkowski billiards inside a unit disc of Minkowski (not necessarily symmetric) norm which is invariant under a linear map of order $k\ge 3$. We find a criterion for the existence of an invariant curve of $k$-periodic orbits. As an application, we get rigidity results for all those billiards.