On $n$-norm preservers and the Aleksandrov conservative $n$-distance problem

TL;DR

This paper strengthens existing results on $n$-norm preservers by showing that transformations preserving $n$-norms or unit $n$-distances are automatically affine or $n$-isometries, under relaxed assumptions.

Contribution

It proves that $n$-norm preserving transformations are automatically plus-minus linear for $n \,\geq\, 3$ and solves an Aleksandrov-type problem in $n$-normed spaces.

Findings

Any transformation preserving the $n$-norm of vectors is plus-minus linear for $n \geq 3.

Surjective transformations preserving unit $n$-distance are $n$-isometries.

Results hold under significantly relaxed assumptions.

Abstract

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.