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 Rⁿ 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 C¹-smoothness. In addition we prove the semi-local and local versions of the result