Billiard characterization of spheres

Abstract

In this note 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 \cite{B0}.

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References in corpus (3)

• Capillary floating and the billiard ball problem
• On recurrence and ergodicity for geodesic flows on noncompact periodic polygonal surfaces
• Addendum to: Capillary floating and the billiard ball problem