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

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 the Aleksandrov problem is solved in n-normed spaces with only the surjectivity 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.