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

2016/09/20 by Xujian Huang, Huang, Xujian, Dongni Tan +1 · 1 citation
Computer Science · Mathematics · #46A03 (Primary) #51K05 (Secondary) #FOS: Mathematics #Fixed Point Theorems Analysis #Functional Analysis (math.FA) #Functional Equations Stability Results #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.1609.06033

openalex publication_date 2016/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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