Gutkin billiard tables in higher dimensions and rigidity

Gutkin found a remarkable class of convex billiard tables in a plane that has a constant angle invariant curve. In this paper we prove that in dimension 3 only a round sphere has such a property. For dimensions greater than 3, a hypersurface with a Gutkin property different from a round sphere, if it exists, must be of constant width and, moreover, it must have very special geometric properties. In the 2D case we prove a rigidity result for Gutkin billiard tables. This is done with the help of a new generating function introduced recently for billiards in our joint paper with Mironov. In the present paper a formula for the generating function in higher dimensions is found.