On configuration spaces of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards

The classical billiard system describes the motion of a point in a plane domain subject to the elastic reflection off the boundary, described by the familiar law of geometrical optics: the angle of incidence equals the angle of reflection; see, e.g., [13, 14] for surveys of mathematical billiards. For every n ≥ 2, the billiard system inside a circle has a very special property: every point of the circle is the starting point of an n-periodic billiard orbit; this orbit is an inscribed regular n-gon. Likewise, an ellipse enjoys the same property for every n ≥ 3; the orbits are inscribed n-gons of extremal perimeter length. A billiard table of constant width also has the property that every point on its boundary belongs to a 2-periodic, back-andforth, billiard trajectory, and this is a dynamic characterization of the curves of constant width. The phase space of the billiard ball map is a cylinder, and the family of n-periodic orbits forms an invariant circle of the billiard ball map consisting of n-periodic points. How exceptional is this property? More specifically, given n ≥ 3, one wants to describe plane billiards such that the billiard ball map has an invariant curve consisting of n-periodic points. The first result in this direc-