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Billiard characterization of spheres

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Abstract

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).

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Acknowledgements

This research was supported in part by ISF grant 162/15. It is a pleasure to thank Yurii Dmitrievich Burago for useful consultations. I am grateful to the anonymous referee for careful reading of the manuscript and improving suggestions.

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  1. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

    Misha Bialy

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  1. Misha Bialy
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Correspondence to Misha Bialy.

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Communicated by Jean-Yves Welschinger.

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Partially supported in part by the Israel Science Foundation Grant 162/15.

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Bialy, M. Billiard characterization of spheres. Math. Ann. 374, 1353–1370 (2019). https://doi.org/10.1007/s00208-019-01831-6

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  • DOI: https://doi.org/10.1007/s00208-019-01831-6

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