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

A. A. Glutsyuk^abc, V. S. Matveev<sup>d

a</sup> Higher School of Contemporary Mathematics, Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, Russia
^b CNRS, UMR 5669 (UMPA, ENS de Lyon), Lyon, France
^c National Research University Higher School of Economics, Moscow, Russia
^d Institute of Mathematics, Friedrich Schiller University of Jena, Jena, Germany

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.
Bibliography: 15 titles.

Keywords: billiard, Minkowski billiard, projective billiard, Binet–Legendre metric.

MSC: 37C83, 37D40, 53B40

Received: 29.08.2024 and 15.02.2025

DOI: 10.4213/sm10182

 English version: Sbornik: Mathematics, 2025, 216:5, 638–653

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