If a Minkowski billiard is projective, then it is the standard billiard

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Abstract

In the recent paper [5] the first-named author proved that if a billiard in a convex domain in $\mathbb R^n$ is simultaneously projective and Minkowski, then it is the standard Euclidean billiard in an appropriate Euclidean structure. The proof was quite complicated and required high smoothness. Here we present a direct simple proof of this result which works in $C^1$-smoothness. In addition, we prove the semi-local and local versions of this result.

About the authors

Alexey Antonovich Glutsyuk

Higher School of Contemporary Mathematics, Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, Russia; CNRS, UMR 5669 (UMPA, ENS de Lyon), Lyon, France; National Research University Higher School of Economics, Moscow, Russia

Author for correspondence.
Email: aglutsyu@ens-lyon.fr
Doctor of physico-mathematical sciences, no status

Vladimir Sergeevich Matveev

Institute of Mathematics, Friedrich Schiller University of Jena, Jena, Germany

Email: vladimir.matveev@uni-jena.de
Candidate of physico-mathematical sciences, Professor

References

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