Billiard characterization of spheres

2018/07/20 by Bialy, Michael
#Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1807.07712

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 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 \citeB0.

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