Mappings of preserving $n$-distance one in $n$-normed spaces

Abstract: We give a positive answer to the Aleksandrov problem in $n$-normed spaces under the surjectivity assumption. Namely, we show that every surjective mapping preserving $n$-distance one is affine, and thus is an $n$-isometry. This is the first time to solve the Aleksandrov problem in $n$-normed spaces with only surjective assumption even in the usual case $n=2$. Finally, when the target space is $n$-strictly convex, we prove that every mapping preserving two $n$-distances with an integer ratio is an affine $n$-isometry.