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

TL;DR

This paper provides a simple, direct proof that a convex billiard in Euclidean space that is both projective and Minkowski must be the standard Euclidean billiard, extending the result to lower smoothness classes.

Contribution

It offers a straightforward proof of a characterization of Euclidean billiards among Minkowski and projective billiards, applicable in $C^1$-smooth settings.

Findings

The proof works in $C^1$-smoothness, reducing smoothness requirements.

The paper establishes semi-local and local versions of the main result.

It confirms that projective Minkowski billiards are necessarily Euclidean.

Abstract

In the recent paper arXiv:2405.13258, the first author of this note 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 the result