Billiard characterization of spheres

TL;DR

This paper proves that in higher dimensions, the only convex hypersurfaces satisfying the Gutkin property in billiards are round spheres, extending previous results to dimensions three and above.

Contribution

The paper establishes that the only convex hypersurface with the Gutkin property in dimensions three and higher is a sphere, generalizing earlier findings.

Findings

Round spheres are uniquely characterized by the Gutkin property in higher dimensions.

The Gutkin property imposes strong geometric constraints leading to spherical symmetry.

Previous results in lower dimensions are extended to all dimensions d ≥ 3.

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 $\mathbf{R}^d, d\geq 3$ is a round sphere. This extends previous results on Gutkin billiards obtained in \cite{B0}.