2003/01/14 by Apoloniusz Tyszka
Mathematics · #Advanced Differential Equations and Dynamical Systems #Functional Equations Stability Results #Nonlinear Differential Equations Analysis #math.MG #msc:51M05
paper · pdf · doi:10.1007/s00010-003-2719-1
published as Aequationes Mathematicae 67 (2004), pp.225-235 · 13 pages, will appear in Aequationes Mathematicae
arxiv created 2003/01/14 · openalex publication_date 2004/06/01 · arxiv updated 2009/11/30 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/29
Let G: Cn × Cn -> C, G((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 G(x,y)=d2 implies G(f(x),f(y))=d2. Let A(n) denote the set of all positive numbers d such that any map f: Rn -> Cn that preserves unit distance preserves also distance d. Let D(n) denote the set of all positive numbers d with the property: if x,y ∈ Rn and |x-y|=d then there exists a finite set S(x,y) with x,y ⊆ S(x,y) ⊆ Rn such that any map f:S(x,y)->Cn that preserves unit distance preserves also the distance between x and y. We prove: (1) A(n) ⊆ d>0: d2 ∈ Q, (2) for n>=2 D(n) is a dense subset of (0,∞). Item (2) implies that each continuous mapping f from Rn to Cn (n>=2) preserving unit distance preserves all distances.