E. Gutkin found a remarkable class of convex billiard tables in the planewhich have a constant angle invariant curve. In this paper we prove that indimension 3 only round sphere has such a property. For dimension greater than 3it must be either a sphere or to have a very special geometric properties. In2-dimensional case we prove a rigidity result for Gutkin billiard tables. Thisis done with the help of a new generating function introduced recently forbilliards in our joint paper with A.E. Mironov. A formula for this generatingfunction in higher dimensions is found.
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