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Billiard characterization of spheres

  • Misha BialyEmail author
Article
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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 [pq] 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).

Notes

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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

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