Integrable outer billiards and rigidity

In the present paper, we introduce a new generating function for outer billiards in the plane. Using this generating function, we prove the following rigidity result: if the exterior of the smooth convex plane curve γ of positive curvature is foliated by continuous curves which are invariant under the outer billiard map, then the curve γ must be an ellipse. In addition to the new generating function used in the proof, we also overcome the non-compactness of the phase space by finding suitable weights in the integral-geometric part of the proof. Thus, we reduce the result to the Blaschke–Santalo inequality.