  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
