Billiard characterization of spheres

Misha Bialy

doi:10.1007/s00208-019-01831-6

Billiard characterization of spheres

Misha Bialy

School of Mathematical Sciences

Tel Aviv University

Research output: Contribution to journal › Article › peer-review

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 δ with the tangent hyperplane at p has the same angle δ with the tangent hyperplane at q. Our main result is that the only convex hypersurface with this property in R^d, d≥ 3 is a round sphere. This extends previous results on Gutkin billiards obtained in Bialy (Nonlinearity 31(5):2281–2293, 2018).

Original language	English
Pages (from-to)	1353-1370
Number of pages	18
Journal	Mathematische Annalen
Volume	374
Issue number	3-4
DOIs	https://doi.org/10.1007/s00208-019-01831-6
State	Published - 6 Aug 2019

All Science Journal Classification (ASJC) codes

General Mathematics

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title = "Billiard characterization of spheres",

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 δ with the tangent hyperplane at p has the same angle δ with the tangent hyperplane at q. Our main result is that the only convex hypersurface with this property in Rd, d≥ 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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year = "2019",

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AB - 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 δ with the tangent hyperplane at p has the same angle δ with the tangent hyperplane at q. Our main result is that the only convex hypersurface with this property in Rd, d≥ 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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DO - 10.1007/s00208-019-01831-6

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EP - 1370

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 3-4

ER -

Billiard characterization of spheres

Abstract

All Science Journal Classification (ASJC) codes

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