Billiard characterization of spheres

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In this paper we study the higher dimensional convex billiards satisfying the so-called Gutkin property. A convex hypersurface S satisfies this property if any chord [p, q] which forms angle $$\delta $$ with the tangent hyperplane at p has the same angle $$\delta $$ with the tangent hyperplane at q. Our main result is that the only convex hypersurface with this property in $$\mathbf {R}^d, d\ge 3$$ is a round sphere. This extends previous results on Gutkin billiards obtained in Bialy (Nonlinearity 31(5):2281–2293, 2018).