  The goal of this paper is to point out that the results obtained in the
recent papers [7,8,10,11] can be seriously strengthened in the sense that we
can significantly relax the assumptions of the main results so that we still
get the same conclusions. In order to do this first, we prove that for $n \geq
3$ any transformation which preserves the $n$-norm of any $n$ vectors is
automatically plus-minus linear. This will give a re-proof of the well-known
Mazur--Ulam-type result that every $n$-isometry is automatically affine ($n
\geq 2$) which was proven in several papers, e.g. in [9]. Second, following the
work of Rassias and \v{S}emrl [23], we provide the solution of a natural
Aleksandrov-type problem in $n$-normed spaces, namely, we show that every
surjective transformation which preserves the unit $n$-distance in both
directions ($n\geq 2$) is automatically an $n$-isometry.
