2002/04/13 by Apoloniusz Tyszka, Tyszka, Apoloniusz
Mathematics · #51M05 (Primary) #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Functional Equations Stability Results #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #math.MG #msc:51M05
paper · pdf · doi:10.48550/arxiv.math/0204171
9 pages, added proofs of technical lemmas
openalex publication_date 2002/04/13 · arxiv created 2002/04/24 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let ϕ((x1,...,xn),(y1,...,yn))=(x1-y1)2+...+(xn-yn)2. We say that f:Rn -> Cn preserves distance d>=0 if for each x,y ∈ Rn ϕ(x,y)=d2 implies ϕ(f(x),f(y))=d2. We prove that if x,y ∈ Rn (n>=3) and |x-y|=(√(2+2/n))k ⋅ (2/n)l (k,l are non-negative integers) then there exists a finite set x,y ⊆ S(x,y) ⊆ Rn such that each unit-distance preserving mapping from S(x,y) to Cn preserves the distance between x and y. It implies that each continuous map from Rn to Cn (n>=3) preserving unit distance preserves all distances.