Gutkin billiard tables in higher dimensions and rigidity

E. Gutkin found a remarkable class of convex billiard tables in the plane which have a constant angle invariant curve. In this paper we prove that in dimension 3 only round sphere has such a property. For dimension greater than 3 it must be either a sphere or to have a very special geometric properties. In 2-dimensional 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 A.E. Mironov. A formula for this generating function in higher dimensions is found.