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

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