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

TL;DR

This paper proves that surjective maps preserving a specific $n$-distance in $n$-normed spaces are affine and $n$-isometries, solving the Aleksandrov problem under minimal assumptions and extending results to strictly convex spaces.

Contribution

It provides the first solution to the Aleksandrov problem in $n$-normed spaces with only surjectivity, establishing that such distance-preserving maps are affine and isometries.

Findings

Surjective $n$-distance one preserving maps are affine in $n$-normed spaces.

Such maps are $n$-isometries.

In strictly convex spaces, preserving two $n$-distances with an integer ratio implies the map is an affine $n$-isometry.

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.