Mappings of Conservative Distances In -Normed spaces ()

Abstract

We show that any mapping between two real $p$-normed spaces, which preserves the unit distance and the midpoint of segments with distance $2^{p}$, is an isometry. Making use of it, we provide an alternative proof of some known results on the Aleksandrov question in normed spaces and also generalise these known results to $p$-normed spaces.